-
question_answer1)
If a, b, c are any vectors, then the true statement is [RPET 1988]
A)
\[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] done
clear
B)
\[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{a}\] done
clear
C)
\[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=\mathbf{a}\,.\,\mathbf{b}\times \mathbf{a}\,.\,\mathbf{c}\] done
clear
D)
\[\mathbf{a}\,.\,(\mathbf{b}-\mathbf{c})=\mathbf{a}\,.\,\mathbf{b}-\mathbf{a}\,.\,\mathbf{c}\] done
clear
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question_answer2)
If a and b are unit vectors such that \[\mathbf{a}\times \mathbf{b}\] is also a unit vector, then the angle between a and b is
A)
0 done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[\pi \] done
clear
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question_answer3)
The points \[A\,(\mathbf{a}),\,B\,(\mathbf{b}),\,C\,(\mathbf{c})\] will be collinear if
A)
\[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\] done
clear
B)
\[\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}=\mathbf{0}\] done
clear
C)
\[\mathbf{a}\,.\,\mathbf{b}+\mathbf{b}\,.\,\mathbf{c}+\mathbf{c}\,.\,\mathbf{a}=0\] done
clear
D)
None of these done
clear
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question_answer4)
\[(\mathbf{a}-\mathbf{b})\times (\mathbf{a}+\mathbf{b})=\] [MP PET 1987]
A)
\[2(\,\mathbf{a}\times \mathbf{b})\] done
clear
B)
\[\mathbf{a}\times \mathbf{b}\] done
clear
C)
\[{{a}^{2}}-{{b}^{2}}\] done
clear
D)
None of these done
clear
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question_answer5)
If \[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0},\] then which relation is correct [RPET 1985; Roorkee 1981; AIEEE 2002]
A)
\[\mathbf{a}=\mathbf{b}=\mathbf{c}=\mathbf{0}\] done
clear
B)
\[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}\] done
clear
C)
\[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}=\mathbf{c}\times \mathbf{a}\] done
clear
D)
None of these done
clear
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question_answer6)
If q be the angle between the vectors a and b and \[|\mathbf{a}\times \mathbf{b}|\,=\mathbf{a}\,.\,\mathbf{b},\] then \[\theta =\] [RPET 1990; MP PET 1990; UPSEAT 2003]
A)
\[\pi \] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
0 done
clear
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question_answer7)
\[(2\mathbf{a}+3\mathbf{b})\times (5\mathbf{a}+7\mathbf{b})=\] [MP PET 1988]
A)
\[\mathbf{a}\times \mathbf{b}\] done
clear
B)
\[\mathbf{b}\times \mathbf{a}\] done
clear
C)
\[\mathbf{a}+\mathbf{b}\] done
clear
D)
\[7\mathbf{a}+10\mathbf{b}\] done
clear
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question_answer8)
If a and b are two vectors such that a . b = 0 and \[\mathbf{a}\times \mathbf{b}=\mathbf{0},\] then [IIT Screening 1989; MNR 1988; UPSEAT 2000, 01]
A)
a is parallel to b done
clear
B)
a is perpendicular to b done
clear
C)
Either a or b is a null vector done
clear
D)
None of these done
clear
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question_answer9)
The components of a vector a along and perpendicular to the non-zero vector b are respectively [IIT 1988]
A)
\[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|},\,\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{a}|}\] done
clear
B)
\[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{b}|},\,\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{b}|}\] done
clear
C)
\[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|},\,\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|}\] done
clear
D)
\[\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{a}|},\,\frac{|\mathbf{a}\,\times \,\mathbf{b}|}{|\mathbf{b}|}\] done
clear
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question_answer10)
\[|\,(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}\,|\,=\,|\mathbf{a}|\,\,|\mathbf{b}|\,\,|\mathbf{c}|,\] if [MP PET 1994; BIT Ranchi 1990; IIT 1982; AMU 2002]
A)
\[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=0\] done
clear
B)
\[\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}=0\] done
clear
C)
\[\mathbf{c}\,.\,\mathbf{a}=\mathbf{a}\,.\,\mathbf{b}=0\] done
clear
D)
\[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}=0\] done
clear
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question_answer11)
Which of the following is not a property of vectors [MP PET 1987]
A)
\[\mathbf{u}\times \mathbf{v}=\mathbf{v}\times \mathbf{u}\] done
clear
B)
\[\mathbf{u}\,.\,\mathbf{v}=\mathbf{v}\,.\,\mathbf{u}\] done
clear
C)
\[{{(\mathbf{u}\times \mathbf{v})}^{2}}={{\mathbf{u}}^{2}}\,.\,{{\mathbf{v}}^{2}}-{{(\mathbf{u}\,.\,\mathbf{v})}^{2}}\] done
clear
D)
\[{{\mathbf{u}}^{2}}=\,|\mathbf{u}{{|}^{2}}\] done
clear
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question_answer12)
The number of vectors of unit length perpendicular to vectors \[\mathbf{a}=(1,\,\,1,\,\,0)\] and \[\mathbf{b}=(0,\,\,1,\,\,1)\] is [BIT Ranchi 1991; IIT 1987; Kurukshetra CEE 1998; DCE 2000; MP PET 2002]
A)
Three done
clear
B)
One done
clear
C)
Two done
clear
D)
Infinite done
clear
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question_answer13)
If \[\mathbf{a}=(1,\,\,-1,\,\,1)\] and \[\mathbf{c}=(-1,\,\,-1,\,\,0),\] then the vector b satisfying \[\mathbf{a}\times \mathbf{b}=\mathbf{c}\] and \[\mathbf{a}\,\,.\,\,\mathbf{b}=1\] is [MP PET 1989]
A)
(1, 0, 0) done
clear
B)
(0, 0, 1) done
clear
C)
(0, ?1, 0) done
clear
D)
None of these done
clear
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question_answer14)
If \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}\ne 0,\] where a, b and c are coplanar vectors, then for some scalar k [Roorkee 1985; RPET 1997]
A)
\[\mathbf{a}+\mathbf{c}=k\,\mathbf{b}\] done
clear
B)
\[\mathbf{a}+\mathbf{b}=k\,\mathbf{c}\] done
clear
C)
\[\mathbf{b}+\mathbf{c}=k\,\mathbf{a}\] done
clear
D)
None of these done
clear
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question_answer15)
If \[\mathbf{a}\ne \mathbf{0},\,\,\mathbf{b}\ne \mathbf{0},\,\,\mathbf{c}\ne \mathbf{0}\], then true statement is [MP PET 1991]
A)
\[\mathbf{a}\times (\mathbf{b}+\mathbf{c})=(\mathbf{c}+\mathbf{b})\times \mathbf{a}\] done
clear
B)
\[\mathbf{a}\,.\,(\mathbf{b}+\mathbf{c})=-(\mathbf{b}+\mathbf{c})\,.\,\mathbf{a}\] done
clear
C)
\[\mathbf{a}\times (\mathbf{b}-\mathbf{c})=(\mathbf{c}-\mathbf{b})\times \mathbf{a}\] done
clear
D)
\[\mathbf{a}\,.\,(\mathbf{b}-\mathbf{c})=(\mathbf{c}-\mathbf{b})\,.\,\mathbf{a}\] done
clear
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question_answer16)
Let a and b be two non-collinear unit vectors. If \[\mathbf{u}=\mathbf{a}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}\] and \[\mathbf{v}=\mathbf{a}\times \mathbf{b},\] then | v | is [IIT 1999]
A)
| u | done
clear
B)
| u |+| u . a | done
clear
C)
| u |+| u . b | done
clear
D)
| u |+ u . (a+b) done
clear
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question_answer17)
If \[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}\ne 0\] and \[\mathbf{a}+\mathbf{c}\ne 0,\] then [RPET 1999]
A)
\[(\mathbf{a}+\mathbf{c})\,\bot \,\mathbf{b}\] done
clear
B)
\[(\mathbf{a}+\mathbf{c})\,\,|\,\,|\,\,\mathbf{b}\] done
clear
C)
\[\mathbf{a}+\mathbf{c}=\mathbf{b}\] done
clear
D)
None of these done
clear
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question_answer18)
A unit vector perpendicular to the plane determined by the points (1, ? 1, 2), (2, 0, ? 1) and (0, 2, 1) is [IIT 1983; MNR 1984]
A)
\[\pm \,\frac{1}{\sqrt{6}}\,(2\mathbf{i}+\mathbf{j}+\mathbf{k})\] done
clear
B)
\[\frac{1}{\sqrt{6}}\,(\mathbf{i}+2\mathbf{j}+\mathbf{k})\] done
clear
C)
\[\frac{1}{\sqrt{6}}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\] done
clear
D)
\[\frac{1}{\sqrt{6}}\,(2\mathbf{i}-\mathbf{j}-\mathbf{k})\] done
clear
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question_answer19)
If \[\mathbf{a}=2\mathbf{i}+3\mathbf{j}-5\mathbf{k},\,\,\mathbf{b}=m\mathbf{i}+n\mathbf{j}+12\mathbf{k}\] and \[\mathbf{a}\times \mathbf{b}=0,\] then \[(m,\,\,n)=\]
A)
\[\left( -\frac{24}{5},\,\frac{36}{5} \right)\] done
clear
B)
\[\left( \frac{24}{5},\,-\frac{36}{5} \right)\] done
clear
C)
\[\left( -\frac{24}{5},\,-\frac{36}{5} \right)\] done
clear
D)
\[\left( \frac{24}{5},\,\frac{36}{5} \right)\] done
clear
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question_answer20)
A unit vector which is perpendicular to \[\mathbf{i}+2\mathbf{j}-2\mathbf{k}\] and \[-\mathbf{i}+2\mathbf{j}+2\mathbf{k}\] is [MP PET 1992]
A)
\[\frac{1}{\sqrt{5}}\,(2\mathbf{i}-\mathbf{k})\] done
clear
B)
\[\frac{1}{\sqrt{5}}\,(-2\mathbf{i}+\mathbf{k})\] done
clear
C)
\[\frac{1}{\sqrt{5}}\,(2\mathbf{i}+\mathbf{j}+\mathbf{k})\] done
clear
D)
\[\frac{1}{\sqrt{5}}\,(2\mathbf{i}+\mathbf{k})\] done
clear
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question_answer21)
If \[A\,(-1,\,\,2,\,\,3),\,\,B\,(1,\,\,1,\,\,1)\] and \[C\,(2,\,\,-1,\,\,3)\] are points on a plane. A unit normal vector to the plane ABC is [BIT Ranchi 1988]
A)
\[\pm \,\left( \frac{2\mathbf{i}+2\mathbf{j}+\mathbf{k}}{3} \right)\] done
clear
B)
\[\pm \,\left( \frac{2\mathbf{i}-2\mathbf{j}+\mathbf{k}}{3} \right)\] done
clear
C)
\[\pm \,\left( \frac{2\mathbf{i}-2\mathbf{j}-\mathbf{k}}{3} \right)\] done
clear
D)
\[-\,\left( \frac{2\mathbf{i}+2\mathbf{j}+\mathbf{k}}{3} \right)\] done
clear
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question_answer22)
The unit vector perpendicular to the vectors \[6\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}-6\mathbf{j}-2\mathbf{k},\] is [IIT 1989; RPET 1996]
A)
\[\frac{2\mathbf{i}-3\mathbf{j}+6\mathbf{k}}{7}\] done
clear
B)
\[\frac{2\mathbf{i}-3\mathbf{j}-6\mathbf{k}}{7}\] done
clear
C)
\[\frac{2\mathbf{i}+3\mathbf{j}-6\mathbf{k}}{7}\] done
clear
D)
\[\frac{2\mathbf{i}+3\mathbf{j}+6\mathbf{k}}{7}\] done
clear
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question_answer23)
For any two vectors a and b, \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] is equal to [Roorkee 1975, 79, 81, 85]
A)
\[{{a}^{2}}-{{b}^{2}}\] done
clear
B)
\[{{a}^{2}}+{{b}^{2}}\] done
clear
C)
\[{{a}^{2}}{{b}^{2}}-{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}\] done
clear
D)
None of these done
clear
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question_answer24)
The unit vector perpendicular to \[3\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[12\mathbf{i}+5\mathbf{j}-5\mathbf{k},\] is [Roorkee 1979; RPET 1989, 91]
A)
\[\frac{5\mathbf{i}-3\mathbf{j}+9\mathbf{k}}{\sqrt{115}}\] done
clear
B)
\[\frac{5\mathbf{i}+3\mathbf{j}-9\mathbf{k}}{\sqrt{115}}\] done
clear
C)
\[\frac{-5\mathbf{i}+3\mathbf{j}-9\mathbf{k}}{\sqrt{115}}\] done
clear
D)
\[\frac{5\mathbf{i}+3\mathbf{j}+9\mathbf{k}}{\sqrt{115}}\] done
clear
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question_answer25)
The sine of the angle between the two vectors \[3\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[12\mathbf{i}+5\mathbf{j}-5\mathbf{k}\] will be [Roorkee 1978]
A)
\[\frac{\sqrt{115}}{\sqrt{14}\sqrt{194}}\] done
clear
B)
\[\frac{51}{\sqrt{14}\sqrt{144}}\] done
clear
C)
\[\frac{\sqrt{64}}{\sqrt{14}\sqrt{194}}\] done
clear
D)
None of these done
clear
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question_answer26)
For any two vectors a and b, if \[\mathbf{a}\times \mathbf{b}=\mathbf{0},\] then [Roorkee 1984]
A)
\[\mathbf{a}=\mathbf{0}\] done
clear
B)
\[\mathbf{b}=\mathbf{0}\] done
clear
C)
Not parallel done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer27)
If a and b are two vectors, then \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] equals [Roorkee 1975, 79, 81, 85]
A)
\[\left| \,\begin{matrix} \mathbf{a}\,\,.\,\,\mathbf{b} & \mathbf{a}\,\,.\,\,\mathbf{a} \\ \mathbf{b}\,\,.\,\,\mathbf{b} & \mathbf{b}\,\,.\,\,\mathbf{a} \\ \end{matrix}\, \right|\] done
clear
B)
\[\left| \,\begin{matrix} \mathbf{a}\,\,.\,\,\mathbf{a} & \mathbf{a}\,\,.\,\,\mathbf{b} \\ \mathbf{b}\,\,.\,\,\mathbf{a} & \mathbf{b}\,\,.\,\,\mathbf{b} \\ \end{matrix}\, \right|\] done
clear
C)
\[\left| \,\begin{matrix} \mathbf{a}\,\,.\,\,\mathbf{b} \\ \mathbf{b}\,\,.\,\,\mathbf{a} \\ \end{matrix}\, \right|\] done
clear
D)
None of these done
clear
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question_answer28)
For any vectors a, b, c \[\mathbf{a}\times (\mathbf{b}+\mathbf{c})+\mathbf{b}\times (\mathbf{c}+\mathbf{a})+\mathbf{c}\times (\mathbf{a}+\mathbf{b})=\] [Roorkee 1981; Kerala (Engg.) 2002]
A)
0 done
clear
B)
\[\mathbf{a}+\mathbf{b}+\mathbf{c}\] done
clear
C)
\[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done
clear
D)
\[\mathbf{a}\times \mathbf{b}\times \mathbf{c}\] done
clear
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question_answer29)
If \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{a}\,.\,\mathbf{c},\,\,\mathbf{a}\,\times \mathbf{b}=\mathbf{a}\times \mathbf{c}\] and \[\mathbf{a}\ne \mathbf{0},\] then [RPET 1990]
A)
\[\mathbf{b}=\mathbf{0}\] done
clear
B)
\[\mathbf{b}\ne \mathbf{c}\] done
clear
C)
\[\mathbf{b}=\mathbf{c}\] done
clear
D)
None of these done
clear
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question_answer30)
If \[|\mathbf{a}|\,=2,\,\,|\mathbf{b}|\,=5\] and \[|\mathbf{a}\times \mathbf{b}|\,=8,\] then a . b is equal to [AI CBSE 1984; RPET 1991]
A)
0 done
clear
B)
2 done
clear
C)
4 done
clear
D)
6 done
clear
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question_answer31)
If \[|\mathbf{a}\,.\,\mathbf{b}|\,=3\] and \[|\mathbf{a}\times \mathbf{b}|\,=4,\] then the angle between a and b is
A)
\[{{\cos }^{-1}}\frac{3}{4}\] done
clear
B)
\[{{\cos }^{-1}}\frac{3}{5}\] done
clear
C)
\[{{\cos }^{-1}}\frac{4}{5}\] done
clear
D)
\[\frac{\pi }{4}\] done
clear
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question_answer32)
If \[\mathbf{a}=2\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=6\mathbf{i}-3\mathbf{j}+2\mathbf{k},\] then the value of \[\mathbf{a}\times \mathbf{b}\] is [MNR 1978; RPET 2001]
A)
\[2\mathbf{i}+2\mathbf{j}-\mathbf{k}\] done
clear
B)
\[6\mathbf{i}-3\mathbf{j}+2\mathbf{k}\] done
clear
C)
\[\mathbf{i}-10\mathbf{j}-18\mathbf{k}\] done
clear
D)
\[\mathbf{i}+\mathbf{j}+\mathbf{k}\] done
clear
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question_answer33)
The scalars l and m such that \[l\mathbf{a}+m\mathbf{b}=\mathbf{c},\] where a, b and c are given vectors, are equal to
A)
\[l=\frac{(\mathbf{c}\times \mathbf{b})\,.\,(\mathbf{a}\times \mathbf{b})}{{{(\mathbf{a}\times \mathbf{b})}^{2}}},\,\,m=\frac{(\mathbf{c}\times \mathbf{a})\,.\,(\mathbf{b}\times \mathbf{a})}{{{(\mathbf{b}\times \mathbf{a})}^{2}}}\] done
clear
B)
\[l=\frac{(\mathbf{c}\times \mathbf{b})\,.\,(\mathbf{a}\times \mathbf{b})}{(\mathbf{a}\times \mathbf{b})},\,\,m=\frac{(\mathbf{c}\times \mathbf{a})\,.\,(\mathbf{b}\times \mathbf{a})}{(\mathbf{b}\times \mathbf{a})}\] done
clear
C)
\[l=\frac{(\mathbf{c}\times \mathbf{b})\,\times \,(\mathbf{a}\times \mathbf{b})}{{{(\mathbf{a}\times \mathbf{b})}^{2}}},\,\,m=\frac{(\mathbf{c}\times \mathbf{a})\,\times \,(\mathbf{b}\times \mathbf{a})}{(\mathbf{b}\times \mathbf{a})}\] done
clear
D)
None of these done
clear
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question_answer34)
\[|\mathbf{a}\times \mathbf{i}{{|}^{2}}+|\mathbf{a}\times \mathbf{j}{{|}^{2}}+|\mathbf{a}\times \mathbf{k}{{|}^{2}}=\] [EAMCET 1988; MP PET 1994, 2004; RPET 2000; Pb. CET 2001; Orissa JEE 2003; AIEEE 2005]
A)
\[|\mathbf{a}{{|}^{2}}\] done
clear
B)
\[2\,\,|\mathbf{a}{{|}^{2}}\] done
clear
C)
\[3\,\,|\mathbf{a}{{|}^{2}}\] done
clear
D)
\[4\,\,|\mathbf{a}{{|}^{2}}\] done
clear
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question_answer35)
A unit vector perpendicular to the plane determined by the points \[P\,(1,\,\,-1,\,\,2),\,\,Q\,(2,\,\,0,\,-1)\] and \[R\,(0,\,\,2,\,\,1)\] is [IIT 1994]
A)
\[\frac{2\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{6}}\] done
clear
B)
\[\frac{2\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{6}}\] done
clear
C)
\[\frac{-2\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{6}}\] done
clear
D)
\[\frac{2\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{6}}\] done
clear
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question_answer36)
A unit vector perpendicular to the vector \[4\mathbf{i}-\mathbf{j}+3\mathbf{k}\] and \[-2\mathbf{i}+\mathbf{j}-2\mathbf{k}\] is [MNR 1995]
A)
\[\frac{1}{3}\,(\mathbf{i}-2\mathbf{j}+2\mathbf{k})\] done
clear
B)
\[\frac{1}{3}\,(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})\] done
clear
C)
\[\frac{1}{3}\,(2\mathbf{i}+\mathbf{j}+2\mathbf{k})\] done
clear
D)
\[\frac{1}{3}\,(2\mathbf{i}-2\mathbf{j}+2\mathbf{k})\] done
clear
View Solution play_arrow
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question_answer37)
Given \[\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=-\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=-\mathbf{i}+2\mathbf{j}-\mathbf{k}.\] A unit vector perpendicular to both \[\mathbf{a}+\mathbf{b}\] and \[\mathbf{b}+\mathbf{c}\] is [Karnataka CET 1993]
A)
i done
clear
B)
j done
clear
C)
k done
clear
D)
\[\frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done
clear
View Solution play_arrow
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question_answer38)
The vectors \[\mathbf{c},\,\,\,\mathbf{a}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\] and \[\mathbf{b}=\mathbf{j}\] are such that a, c, b form a right handed system, then c is [DCE 1999]
A)
\[z\mathbf{i}-x\mathbf{k}\] done
clear
B)
0 done
clear
C)
\[y\mathbf{j}\] done
clear
D)
\[-z\mathbf{i}+x\mathbf{k}\] done
clear
View Solution play_arrow
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question_answer39)
If A, B, C, D are any four points in space, then \[|\overrightarrow{AB}\times \overrightarrow{CD}+\overrightarrow{BC}\times \overrightarrow{AD}+\overrightarrow{CA}\times \overrightarrow{BD}|\] is equal to
A)
\[2\Delta \] done
clear
B)
\[4\Delta \] done
clear
C)
\[3\Delta \] done
clear
D)
\[5\Delta \] (where D denotes the area of \[\Delta ABC\]) done
clear
View Solution play_arrow
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question_answer40)
If \[{{(\mathbf{a}\times \mathbf{b})}^{2}}+{{(\mathbf{a}\,\,.\,\,\mathbf{b})}^{2}}=144\] and \[|\mathbf{a}|\,=4,\] then \[|\mathbf{b}|\,=\] [EAMCET 1994]
A)
16 done
clear
B)
8 done
clear
C)
3 done
clear
D)
12 done
clear
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question_answer41)
\[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a};\,\,\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b};\,\,\mathbf{a}\ne 0;\,\,\mathbf{b}\ne 0;\,\,\mathbf{a}\ne \lambda \mathbf{b},\,\,\]a is not perpendicular to b, then \[\mathbf{r}=\] [EAMCET 1993]
A)
\[\mathbf{a}-\mathbf{b}\] done
clear
B)
\[\mathbf{a}+\mathbf{b}\] done
clear
C)
\[\mathbf{a}\times \mathbf{b}+\mathbf{a}\] done
clear
D)
\[\mathbf{a}\times \mathbf{b}+\mathbf{b}\] done
clear
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question_answer42)
If \[\mathbf{i},\,\,\mathbf{j},\,\,\mathbf{k}\] are unit orthonormal vectors and a is a vector, if \[\mathbf{a}\times \mathbf{r}=\mathbf{j},\] then a . r is [EAMCET 1990]
A)
0 done
clear
B)
1 done
clear
C)
? 1 done
clear
D)
Arbitrary scalar done
clear
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question_answer43)
A unit vector perpendicular to each of the vector \[2\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[3\mathbf{i}+4\mathbf{j}-\mathbf{k}\] is equal to [MP PET 2003]
A)
\[\frac{(-3\mathbf{i}+5\mathbf{j}+11\mathbf{k})}{\sqrt{155}}\] done
clear
B)
\[\frac{(3\mathbf{i}-5\mathbf{j}+11\mathbf{k})}{\sqrt{155}}\] done
clear
C)
\[\frac{(6\mathbf{i}-4\mathbf{j}-\mathbf{k})}{\sqrt{53}}\] done
clear
D)
\[\frac{(5\mathbf{i}+3\mathbf{j})}{\sqrt{34}}\] done
clear
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question_answer44)
If \[\overrightarrow{A}=3\mathbf{i}+\mathbf{j}+2\mathbf{k}\] and \[\overrightarrow{B}=2\mathbf{i}-2\mathbf{j}+4\mathbf{k}\] and q is the angle between \[\overrightarrow{A}\] and \[\overrightarrow{B},\] then the value of \[\sin \theta \] is
A)
\[\frac{2}{\sqrt{7}}\] done
clear
B)
\[\sqrt{\frac{2}{7}}\] done
clear
C)
\[\frac{4}{\sqrt{7}}\] done
clear
D)
\[\frac{3}{\sqrt{7}}\] done
clear
View Solution play_arrow
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question_answer45)
A unit vector perpendicular to vector c and coplanar with vectors a and b is [MP PET 1999]
A)
\[\frac{\mathbf{a}\times (\mathbf{b}\times \mathbf{c})}{|\mathbf{a}\times (\mathbf{b}\times \mathbf{c})|}\] done
clear
B)
\[\frac{\mathbf{b}\times (\mathbf{c}\times \mathbf{a})}{|\mathbf{b}\times (\mathbf{c}\times \mathbf{a})|}\] done
clear
C)
\[\frac{\mathbf{c}\times (\mathbf{a}\times \mathbf{b})}{|\mathbf{c}\times (\mathbf{a}\times \mathbf{b})|}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer46)
\[|\mathbf{a}\times \mathbf{b}{{|}^{2}}+\,{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}=\] [MP PET 1989, 97, 2004]
A)
\[(\mathbf{a}\times \mathbf{a})\,.\,(\mathbf{b}\times \mathbf{b})\] done
clear
B)
\[(\mathbf{a}\,.\,\mathbf{a})\,(\mathbf{b}\,.\,\mathbf{b})\] done
clear
C)
\[|\,(\mathbf{a}\times \mathbf{b})\,|\,\,(\mathbf{a}\,.\,\mathbf{b})\] done
clear
D)
\[2\,(\mathbf{a}\,.\,\mathbf{b})\,(\mathbf{a}\,.\,\mathbf{b})\] done
clear
View Solution play_arrow
-
question_answer47)
If the position vectors of three points \[A,B\] and \[C\] are respectively i + j + k, 2i + 3j ? 4k and 7i + 4j + 9k, then the unit vector to the plane containing the triangle \[ABC\] is [DCE 1999]
A)
31i ? 18j ? 9k done
clear
B)
\[\frac{31i-38j-9 k}{\sqrt{2486}}\] done
clear
C)
\[\frac{31i+18j+9k}{\sqrt{2486}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer48)
If a, b, c are position vector of vertices of a triangle \[ABC\], then unit vector perpendicular to its plane is [RPET 1999]
A)
\[a\times b+b\times c+c\times a\] done
clear
B)
\[\frac{a\times b+b\times c+c\times a}{|a\times b+b\times c+c\times a|}\] done
clear
C)
\[\frac{a\times b}{|a\times b|}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer49)
If \[\theta \] is the angle between the vectors a and b, then \[\frac{|a\times b|}{|a\,.\,b|}\] equal to [Karnataka CET 1999]
A)
\[\tan \theta \] done
clear
B)
\[-\tan \theta \] done
clear
C)
\[\cot \theta \] done
clear
D)
\[-\cot \theta \] done
clear
View Solution play_arrow
-
question_answer50)
If the vectors \[\mathbf{a},\,\mathbf{b}\] and c are represented by the sides \[BC,\,CA\] and \[AB\] respectively of the \[\Delta ABC\], then [IIT Screening 2000]
A)
\[\mathbf{a}\,\mathbf{.}\,\mathbf{b}+\mathbf{b}\,\mathbf{.}\,\mathbf{c}+\mathbf{c}\,\mathbf{.}\,\mathbf{a}=0\] done
clear
B)
\[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}=\mathbf{c}\times \mathbf{a}\] done
clear
C)
\[\mathbf{a}\,\mathbf{.}\,\mathbf{b}=\mathbf{b}\,\mathbf{.}\,\mathbf{c}=\mathbf{c}\,\mathbf{.}\,\mathbf{a}\] done
clear
D)
\[\mathbf{a}\times \mathbf{b}=\mathbf{b}\times \mathbf{c}=\mathbf{c}\times \mathbf{a}=0\] done
clear
View Solution play_arrow
-
question_answer51)
A vector perpendicular to both of the vectors \[i+j+k\] and \[i+j\] is [RPET 2000]
A)
i + j done
clear
B)
i ? j done
clear
C)
\[c(i-j)\], c is a scalar done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer52)
A unit vector perpendicular to the plane of \[a=2i-6j-3k\], \[b=4i+3j-k\] is [MP PET 2000]
A)
\[\frac{4i+3j-k}{\sqrt{26}}\] done
clear
B)
\[\frac{2i-6j-3k}{7}\] done
clear
C)
\[\frac{3i-2j+6k}{7}\] done
clear
D)
\[\frac{2i-3j-6k}{7}\] done
clear
View Solution play_arrow
-
question_answer53)
The unit vector perpendicular to both the vectors i ? 2j + 3k and i + 2j ? k is [DCE 2001]
A)
\[\frac{1}{\sqrt{3}}(-i+j+k)\] done
clear
B)
\[(-\mathbf{i}+\mathbf{j}+\mathbf{k})\] done
clear
C)
\[\frac{(i+j-k)}{\sqrt{3}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer54)
The unit vector perpendicular to the vectors \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[2\mathbf{i}+3\mathbf{j}-\mathbf{k}\] is [Karnataka CET 2001]
A)
\[\frac{-2i+3j+5k}{\sqrt{30}}\] done
clear
B)
\[\frac{-2\mathbf{i}+5\mathbf{j}+6\mathbf{k}}{\sqrt{38}}\]\[\] done
clear
C)
\[\frac{-2i+3j+5k}{\sqrt{38}}\] done
clear
D)
\[\frac{-2i+4j+5k}{\sqrt{38}}\] done
clear
View Solution play_arrow
-
question_answer55)
If \[\mathbf{a}=2\mathbf{i}-3\,\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}+4\mathbf{j}-2\mathbf{k}\], then \[\mathbf{a}\times \mathbf{b}\] is [MP PET 2001]
A)
10i + 2j + 11k done
clear
B)
10i + 3j + 11k done
clear
C)
10i ? 3j + 11k done
clear
D)
10i ? 3j ? 10k done
clear
View Solution play_arrow
-
question_answer56)
If \[|\mathbf{a}|\,=4,\,|\mathbf{b}|\,=2\] and the angle between a and b is \[\frac{\pi }{6}\], then \[{{(\mathbf{a}\times \mathbf{b})}^{2}}\] is equal to [AIEEE 2002]
A)
48 done
clear
B)
16 done
clear
C)
8 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer57)
If \[\mathbf{a}=2\mathbf{i}+4\mathbf{j}-5\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}\], then \[|\mathbf{a}\times \mathbf{b}|\] is [UPSEAT 2002]
A)
\[11\sqrt{5}\] done
clear
B)
\[11\sqrt{3}\] done
clear
C)
\[11\sqrt{7}\] done
clear
D)
\[11\sqrt{2}\] done
clear
View Solution play_arrow
-
question_answer58)
The unit vector perpendicular to both \[\mathbf{i}+\mathbf{j}\] and \[\mathbf{j}+\mathbf{k}\] is [Kerala (Engg.) 2002]
A)
i ? j + k done
clear
B)
i + j + k done
clear
C)
\[\frac{\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{3}}\] done
clear
D)
\[\frac{\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done
clear
View Solution play_arrow
-
question_answer59)
A unit vector in the plane of the vectors \[2\mathbf{i}+\mathbf{j}+\mathbf{k},\,\] \[\,\mathbf{i}-\mathbf{j}+\mathbf{k}\] and orthogonal to \[5\mathbf{i}+2\mathbf{j}+6\mathbf{k}\] is [IIT Screening 2004]
A)
\[\frac{6\mathbf{i}-5\mathbf{k}}{\sqrt{61}}\] done
clear
B)
\[\frac{3\mathbf{j}-\mathbf{k}}{\sqrt{10}}\] done
clear
C)
\[\frac{2\mathbf{i}-5\mathbf{j}}{\sqrt{29}}\] done
clear
D)
\[\frac{2\mathbf{i}+\mathbf{j}-2\mathbf{k}}{3}\] done
clear
View Solution play_arrow
-
question_answer60)
Let \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] be three vectors such that \[\mathbf{a}\ne 0,\] and \[\mathbf{a}\times \mathbf{b}=2\mathbf{a}\times \mathbf{c},\,\,|\mathbf{a}|\,=\,|\mathbf{c}|\,=\,1,\,|\mathbf{b}|\,=4\] and \[|\mathbf{b}\times \mathbf{c}|\,=15.\] If \[\mathbf{b}-2\mathbf{c}=\lambda \mathbf{a},\] then l equals to [Orissa JEE 2004]
A)
1 done
clear
B)
\[\pm \,4\] done
clear
C)
3 done
clear
D)
? 2 done
clear
View Solution play_arrow
-
question_answer61)
The area of a triangle whose vertices are \[A\,(1,\,-1,\,2),\] \[B\,(2,\,1,\,-1)\] and \[C\,(3,\,-1,\,2)\] is [MNR 1983; IIT 1983]
A)
13 done
clear
B)
\[\sqrt{13}\] done
clear
C)
6 done
clear
D)
\[\sqrt{6}\] done
clear
View Solution play_arrow
-
question_answer62)
If vertices of a triangle are \[A(1,\,-1,\,2),\,B(2,\,0,\,-1)\] and \[C(0,\,2,\,1),\] then the area of a triangle is [RPET 2000]
A)
\[\sqrt{6}\] done
clear
B)
\[2\sqrt{6}\] done
clear
C)
\[3\sqrt{6}\] done
clear
D)
\[4\sqrt{6}\] done
clear
View Solution play_arrow
-
question_answer63)
The area of triangle whose vertices are \[(1,\,2,\,3),\,(2,\,5,\,-1)\] and \[(-1,\,1,\,2)\] is [Kerala (Engg.) 2002]
A)
150 sq. unit done
clear
B)
145 sq. unit done
clear
C)
\[\frac{\sqrt{155}}{2}\] sq. unit done
clear
D)
\[\frac{155}{2}\] sq. unit done
clear
View Solution play_arrow
-
question_answer64)
The area of a parallelogram whose two adjacent sides are represented by the vector \[3\mathbf{i}-\mathbf{k}\] and \[\mathbf{i}+2\mathbf{j}\] is [MNR 1981]
A)
\[\frac{1}{2}\sqrt{17}\] done
clear
B)
\[\frac{1}{2}\sqrt{14}\] done
clear
C)
\[\sqrt{41}\] done
clear
D)
\[\frac{1}{2}\sqrt{7}\] done
clear
View Solution play_arrow
-
question_answer65)
The area of the parallelogram whose diagonals are \[\mathbf{a}=3\,\mathbf{i}+\mathbf{j}-2\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}-3\,\mathbf{j}+4\,\mathbf{k}\] is [MP PET 1988, 93; MNR 1985]
A)
\[10\sqrt{3}\] done
clear
B)
\[5\sqrt{3}\] done
clear
C)
8 done
clear
D)
4 done
clear
View Solution play_arrow
-
question_answer66)
The position vectors of the points A, B and C are \[\mathbf{i}+\mathbf{j},\,\,\mathbf{j}+\mathbf{k}\] and \[\mathbf{k}+\mathbf{i}\] respectively. The vector area of the \[\Delta ABC=\pm \,\frac{1}{2}\overrightarrow{\alpha }\] where \[\overrightarrow{\alpha }=\] [MP PET 1989]
A)
\[-\mathbf{i}+\mathbf{j}+\mathbf{k}\] done
clear
B)
\[\mathbf{i}-\mathbf{j}+\mathbf{k}\] done
clear
C)
\[\mathbf{i}+\mathbf{j}-\mathbf{k}\] done
clear
D)
\[\mathbf{i}+\mathbf{j}+\mathbf{k}\] done
clear
View Solution play_arrow
-
question_answer67)
If \[\overrightarrow{OA}=3\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\overrightarrow{OB}=\mathbf{i}+3\mathbf{j}+\mathbf{k}\], then the area of the triangle OAB is
A)
\[\sqrt{15}\] done
clear
B)
\[3\sqrt{5}\] done
clear
C)
\[\frac{3}{2}\sqrt{10}\] done
clear
D)
\[\frac{5\sqrt{5}}{3}\] done
clear
View Solution play_arrow
-
question_answer68)
Let a, b, c be the position vectors of the vertices of a triangle ABC. The vector area of triangle ABC is [MP PET 1990; EAMCET 2003]
A)
\[\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}\] done
clear
B)
\[\frac{1}{4}(\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a})\] done
clear
C)
\[\frac{1}{2}(\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a})\] done
clear
D)
\[\mathbf{b}\times \mathbf{a}+\mathbf{c}\times \mathbf{b}+\mathbf{a}\times \mathbf{c}\] done
clear
View Solution play_arrow
-
question_answer69)
If \[|\mathbf{a}|\,=2,\,\,|\mathbf{b}|\,=3\] and a, b are mutually perpendicular, then the area of the triangle whose vertices are \[\mathbf{0},\,\,\mathbf{a}+\mathbf{b},\,\,\mathbf{a}-\mathbf{b}\] is
A)
5 done
clear
B)
1 done
clear
C)
6 done
clear
D)
8 done
clear
View Solution play_arrow
-
question_answer70)
If \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}-2\mathbf{j}+\mathbf{k}\] represents the adjacent sides of a parallelogram, then the area of this parallelogram is [Roorkee 1978, 79; MP PET 1990; RPET 1988, 89, 91]
A)
\[4\sqrt{3}\] done
clear
B)
\[6\sqrt{3}\] done
clear
C)
\[8\sqrt{3}\] done
clear
D)
\[16\sqrt{3}\] done
clear
View Solution play_arrow
-
question_answer71)
If \[3\mathbf{i}+4\mathbf{j}\] and \[-5\mathbf{i}+7\mathbf{j}\] are the vector sides of any triangle, then its area is given by [RPET 1987, 90]
A)
41 done
clear
B)
47 done
clear
C)
\[\frac{41}{2}\] done
clear
D)
\[\frac{47}{2}\] done
clear
View Solution play_arrow
-
question_answer72)
If the vectors \[\mathbf{i}-3\mathbf{j}+2\mathbf{k}\], \[-\mathbf{i}+2\mathbf{j}\] represents the diagonals of a parallelogram, then its area will be [Roorkee 1976]
A)
\[\sqrt{21}\] done
clear
B)
\[\frac{\sqrt{21}}{2}\] done
clear
C)
\[2\sqrt{21}\] done
clear
D)
\[\frac{\sqrt{21}}{4}\] done
clear
View Solution play_arrow
-
question_answer73)
The area of the parallelogram whose diagonals are the vectors \[2\mathbf{a}-\mathbf{b}\] and \[4\mathbf{a}-5\mathbf{b},\] where a and b are the unit vectors forming an angle of \[{{45}^{o}},\] is
A)
\[3\sqrt{2}\] done
clear
B)
\[\frac{3}{\sqrt{2}}\] done
clear
C)
\[\sqrt{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer74)
The area of a parallelogram whose adjacent sides are \[\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[2\mathbf{i}+\mathbf{j}-4\mathbf{k},\] is [MP PET 1996, 2000]
A)
\[5\sqrt{3}\] done
clear
B)
\[10\sqrt{3}\] done
clear
C)
\[5\sqrt{6}\] done
clear
D)
\[10\sqrt{6}\] done
clear
View Solution play_arrow
-
question_answer75)
If the diagonals of a parallelogram are represented by the vectors \[3\mathbf{i}+\mathbf{j}-2\mathbf{k}\] and \[\mathbf{i}+3\mathbf{j}-4\mathbf{k},\] then its area in square unit is [MP PET 1998]
A)
\[5\sqrt{3}\] done
clear
B)
\[6\sqrt{3}\] done
clear
C)
\[\sqrt{26}\] done
clear
D)
\[\sqrt{42}\] done
clear
View Solution play_arrow
-
question_answer76)
The area of a parallelogram whose adjacent sides are given by the vectors \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[-3\mathbf{i}-2\mathbf{j}+\mathbf{k}\] (in square unit) is [Karnataka CET 2001; Pb. CET 2004]
A)
\[\sqrt{180}\] done
clear
B)
\[\sqrt{140}\] done
clear
C)
\[\sqrt{80}\] done
clear
D)
\[\sqrt{40}\] done
clear
View Solution play_arrow
-
question_answer77)
If \[\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k},\,\mathbf{b}=\mathbf{i}+3\mathbf{j}+5\mathbf{k}\] and \[\mathbf{c}=7\mathbf{i}+9\mathbf{j}+11\mathbf{k}\], then the area of the parallelogram having diagonals a + b and b + c is [Kurukshetra CEE 2002]
A)
\[4\sqrt{6}\] done
clear
B)
\[\frac{1}{2}\sqrt{21}\] done
clear
C)
\[\frac{\sqrt{6}}{2}\] done
clear
D)
\[\sqrt{6}\] done
clear
View Solution play_arrow
-
question_answer78)
The area of the parallelogram whose diagonals are \[\frac{3}{2}\mathbf{i}+\frac{1}{2}\mathbf{j}-\mathbf{k}\] and \[2\mathbf{i}-6\mathbf{j}+8\mathbf{k}\] is [UPSEAT 2002]
A)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=bc+ca+ab\] done
clear
B)
\[5\sqrt{2}\] done
clear
C)
\[({{a}^{2}}+ab,\,{{b}^{2}}+ab,\,-ab)\] done
clear
D)
\[(-bc,\,{{b}^{2}}+bc,\,{{c}^{2}}+bc),\] done
clear
View Solution play_arrow
-
question_answer79)
The area of the triangle having vertices as \[\mathbf{i}-2\mathbf{j}+3\mathbf{k},\] \[\,-2\mathbf{i}+3\mathbf{j}+\mathbf{k}\] , \[4\mathbf{i}-7\mathbf{j}+7\mathbf{k}\] is [MP PET 2004]
A)
26 done
clear
B)
11 done
clear
C)
36 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer80)
The area of the parallelogram whose adjacent sides are \[\mathbf{i}-\mathbf{k}\] and \[2\mathbf{j}+3\mathbf{k}\] is [UPSEAT 2004]
A)
2 done
clear
B)
4 done
clear
C)
\[\sqrt{17}\] done
clear
D)
\[2\sqrt{13}\] done
clear
View Solution play_arrow
-
question_answer81)
The moment of the force \[\overrightarrow{F}\] acting at a point P, about the point C is [MP PET 1987]
A)
\[\overrightarrow{F}\times \overrightarrow{CP}\] done
clear
B)
\[\overrightarrow{CP}\,.\,\overrightarrow{F}\] done
clear
C)
A vector having the same direction as \[\overrightarrow{F}\] done
clear
D)
\[\overrightarrow{CP}\times \overrightarrow{F}\] done
clear
View Solution play_arrow
-
question_answer82)
Three forces \[\mathbf{i}+2\,\mathbf{j}-3\,\mathbf{k},\,\,2\,\mathbf{i}+3\,\mathbf{j}+4\,\mathbf{k}\] and \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] are acting on a particle at the point (0, 1, 2). The magnitude of the moment of the forces about the point \[(1,\,-2,\,0)\] is [MNR 1983]
A)
\[2\sqrt{35}\] done
clear
B)
\[6\sqrt{10}\] done
clear
C)
\[4\sqrt{17}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer83)
Let the points A, B and P be (? 2, 2, 4), (2, 6, 3) and (1,2,1) respectively. The magnitude of the moment of the force represented by \[\overrightarrow{AB}\] and acting at A about P is [MP PET 1987]
A)
15 done
clear
B)
\[3\sqrt{41}\] done
clear
C)
\[3\sqrt{57}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer84)
The moment of a force represented by \[\overrightarrow{F}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] about the point \[2\,\mathbf{i}-\mathbf{j}+\mathbf{k}=\] [BIT Ranchi 1992]
A)
\[5\mathbf{i}-5\mathbf{j}+5\mathbf{k}\] done
clear
B)
\[5\mathbf{i}+5\mathbf{j}-5\mathbf{k}\] done
clear
C)
\[-5\mathbf{i}+5\mathbf{j}+5\mathbf{k}\] done
clear
D)
\[-5\mathbf{i}-5\mathbf{j}+5\mathbf{k}\] done
clear
View Solution play_arrow
-
question_answer85)
A force of magnitude 6 acts along the vector \[(9,\,6,\,-2)\] and passes through a point A (4, ? 1, ?7). The moment of the force about the point O (1, ? 3, 2) is
A)
\[\frac{150}{11}\,(2\mathbf{i}-3\mathbf{j})\] done
clear
B)
\[\frac{6}{11}\,(50\mathbf{i}-75\mathbf{j}+36\mathbf{k})\] done
clear
C)
\[150\,(2\mathbf{i}-3\mathbf{j})\] done
clear
D)
\[6\,(50\mathbf{i}-75\mathbf{j}+36\mathbf{k})\] done
clear
View Solution play_arrow
-
question_answer86)
A force \[\mathbf{F}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] acts at a point A, whose position vector is \[2\mathbf{i}-\mathbf{j}\]. The moment of F about the origin is [Karnataka CET 2000]
A)
\[\mathbf{i}+2\mathbf{j}-4\mathbf{k}\] done
clear
B)
\[\mathbf{i}-2\mathbf{j}-4\mathbf{k}\] done
clear
C)
\[\mathbf{i}+2\mathbf{j}+4\mathbf{k}\] done
clear
D)
\[\mathbf{i}-2\mathbf{j}+4\mathbf{k}\] done
clear
View Solution play_arrow
-
question_answer87)
If \[\mathbf{a}=\mathbf{i}-\mathbf{j},\mathbf{b}=\mathbf{i}+\mathbf{j},\,\,\,\mathbf{c}=\mathbf{i}+3\mathbf{j}+5\mathbf{k}\]and\[\mathbf{n}\]is a unit vector such that \[\mathbf{b}.\mathbf{n}=0,\mathbf{a}.\mathbf{n}=0\]then the value of \[|\mathbf{c}\ .\ \mathbf{n}|\] is equal to [DCE 2005]
A)
1 done
clear
B)
3 done
clear
C)
5 done
clear
D)
2 done
clear
View Solution play_arrow
-
question_answer88)
A unit vector perpendicular to the plane containing the vectors \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[-\mathbf{i}+\mathbf{j}+\mathbf{k}\] is [Karnataka CET 2005]
A)
\[\frac{\mathbf{i}-\mathbf{j}}{\sqrt{2}}\] done
clear
B)
\[\frac{\mathbf{i}+\mathbf{k}}{\sqrt{2}}\] done
clear
C)
\[\frac{\mathbf{j}-\mathbf{k}}{\sqrt{2}}\] done
clear
D)
\[\frac{\mathbf{i}+\mathbf{j}}{\sqrt{2}}\] done
clear
View Solution play_arrow