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question_answer1)
\[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is coplanar with
A)
b and c done
clear
B)
c and a done
clear
C)
a and b done
clear
D)
a, b and c done
clear
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question_answer2)
If \[\mathbf{u}=\mathbf{i}\times (\mathbf{a}\times \mathbf{i})+\mathbf{j}\times (\mathbf{a}\times \mathbf{j})+\mathbf{k}\times (\mathbf{a}\times \mathbf{k}),\]then [RPET 1989, 97; MNR 1986, 93; MP PET 1987, 98, 99, 2004; UPSEAT 2000, 02; Kerala (Engg.) 2002]
A)
\[\mathbf{u}=0\] done
clear
B)
\[\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}\] done
clear
C)
\[\mathbf{u}=2\mathbf{a}\] done
clear
D)
\[\mathbf{u}=\mathbf{a}\] done
clear
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question_answer3)
If \[\mathbf{a}=\mathbf{i}+2\mathbf{j}-2\mathbf{k},\,\,\mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}\]and \[\mathbf{c}=\mathbf{i}+3\mathbf{j}-\mathbf{k},\] then \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is equal to [RPET 1989]
A)
\[20\mathbf{i}-3\mathbf{j}+7\mathbf{k}\] done
clear
B)
\[20\mathbf{i}-3\mathbf{j}-7\mathbf{k}\] done
clear
C)
\[20\mathbf{i}+3\mathbf{j}-7\mathbf{k}\] done
clear
D)
None of these done
clear
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question_answer4)
If \[\mathbf{\alpha }=2\mathbf{i}+3\mathbf{j}-\mathbf{k},\,\,\mathbf{\beta }=-\mathbf{i}+2\mathbf{j}-4\mathbf{k}\] and \[\mathbf{\gamma }=\mathbf{i}+\mathbf{j}+\mathbf{k},\] then \[(\mathbf{\alpha }\times \mathbf{\beta }).(\mathbf{\alpha }\times \mathbf{\gamma })\] is equal to [MNR 1984; UPSEAT 2000; Orissa JEE 2005]
A)
60 done
clear
B)
64 done
clear
C)
74 done
clear
D)
? 74 done
clear
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question_answer5)
If \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\mathbf{0},\] then [RPET 1995]
A)
\[|\mathbf{a}|\,=\,|\mathbf{b}|\,.\,|\mathbf{c}|\,=1\] done
clear
B)
\[\mathbf{b}||\mathbf{c}\] done
clear
C)
\[\mathbf{a}||\mathbf{b}\] done
clear
D)
\[\mathbf{b}\,\bot \,\mathbf{c}\] done
clear
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question_answer6)
\[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is equal to [RPET 1995; Kurukshetra CEE 1998; MP PET 2003]
A)
\[(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{a})\,\mathbf{b}\] done
clear
B)
\[(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{a}-(\mathbf{b}\,.\,\mathbf{c})\,\mathbf{a}\] done
clear
C)
\[(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}\] done
clear
D)
\[(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{c}-(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}\] done
clear
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question_answer7)
If \[\mathbf{a}\times \mathbf{b}=\mathbf{c},\,\,\mathbf{b}\times \mathbf{c}=\mathbf{a}\] and a, b, c be moduli of the vectors a, b, c respectively, then
A)
\[a=1,\,\,b=c\] done
clear
B)
\[c=1,\,\,a=1\] done
clear
C)
\[b=2,\,\,c=2a\] done
clear
D)
\[b=1,\,\,c=a\] done
clear
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question_answer8)
If \[\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k},\] \[\mathbf{b}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}-2\mathbf{j}+2\mathbf{k},\]then \[(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] is equal to
A)
\[24\mathbf{i}+7\mathbf{j}-5\mathbf{k}\] done
clear
B)
\[7\mathbf{i}-24\mathbf{j}+5\mathbf{k}\] done
clear
C)
\[12\mathbf{i}+3\mathbf{j}-5\mathbf{k}\] done
clear
D)
\[\mathbf{i}+\mathbf{j}-7\mathbf{k}\] done
clear
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question_answer9)
\[\mathbf{i}\times (\mathbf{j}\times \mathbf{k})=\] [RPET 1988; MP PET 1997]
A)
1 done
clear
B)
0 done
clear
C)
? 1 done
clear
D)
None of these done
clear
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question_answer10)
If three unit vectors a, b, c are such that \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}}{2},\] then the vector a makes with b and c respectively the angles [MP PET 1998]
A)
\[{{40}^{o}},\,{{80}^{o}}\] done
clear
B)
\[{{45}^{o}},\,\,{{45}^{o}}\] done
clear
C)
\[{{30}^{o}},\,\,{{60}^{o}}\] done
clear
D)
\[{{90}^{o}},\,\,{{60}^{o}}\] done
clear
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question_answer11)
Let \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}\] and \[\mathbf{b}=\mathbf{i}+\mathbf{j}.\] If c is a vector such that \[\mathbf{a}\,.\,\mathbf{c}=\,|\mathbf{c}|,\,\,|\mathbf{c}-\mathbf{a}|\,=2\sqrt{2}\]and the angle between \[(\mathbf{a}\times \mathbf{b})\] and c is \[{{30}^{o}}\], then \[|\,(\mathbf{a}\times \mathbf{b})\times \mathbf{c}|\,=\] [IIT 1999]
A)
\[\frac{2}{3}\] done
clear
B)
\[\frac{3}{2}\] done
clear
C)
2 done
clear
D)
3 done
clear
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question_answer12)
\[i\times (j\times k)+j\times (k\times i)\,+k\times (i\times j)\] equals [RPET 1999]
A)
i done
clear
B)
j done
clear
C)
k done
clear
D)
0 done
clear
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question_answer13)
\[[\mathbf{b}\times \mathbf{c}\,\,\mathbf{c}\times \mathbf{a}\,\,\mathbf{a}\times \mathbf{b}]\] is equal to [MP PET 2004]
A)
\[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] done
clear
B)
\[2\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done
clear
C)
\[{{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}^{2}}\] done
clear
D)
\[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done
clear
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question_answer14)
Given three unit vector \[a,\,b,\,c\] such that \[\mathbf{a}\,\bot \,\mathbf{b}\] and \[a||c\], then \[a\times (b\times c)\] is [AMU 1999]
A)
a done
clear
B)
b done
clear
C)
c done
clear
D)
0 done
clear
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question_answer15)
If \[\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\mathbf{b}=\mathbf{i}-\mathbf{j}+\mathbf{k},\]\[\mathbf{c}=\mathbf{i}-\mathbf{j}-\mathbf{k}\], then \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is [MP PET 2000]
A)
i ? j + k done
clear
B)
2i ? 2j done
clear
C)
3i ? j + k done
clear
D)
2i + 2j ? k done
clear
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question_answer16)
If \[\overrightarrow{A}=i-2j-3k,\,\overrightarrow{B}=2i+j-k,\,\overrightarrow{C}=i+3j-2k\], then \[(\overrightarrow{A}\times \overrightarrow{B})\times \overrightarrow{C}\] is [MP PET 2001]
A)
\[5(-\,i+3j+4k)\] done
clear
B)
\[4(-\,i+3j+4k)\] done
clear
C)
\[5(-\,i-3j-4k)\] done
clear
D)
\[4(i+3j+4k)\] done
clear
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question_answer17)
\[a\times (b\times c)+b\times (c\times a)+c\times (a\times b)=\] [RPET 2003]
A)
0 done
clear
B)
2[a b c] done
clear
C)
a + b + c done
clear
D)
3[a b c] done
clear
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question_answer18)
Let \[a,\,b,\,c\] be three vectors from \[a\times (b\times c)=(a\times b)\times c\], if [Orissa JEE 2003]
A)
\[b\times (a\times c)=0\] done
clear
B)
\[a(b\times c)=0\] done
clear
C)
\[c\times a=a\times b\] done
clear
D)
\[c\times b=b\times a\] done
clear
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question_answer19)
If \[a=i+j+k,\,\,b=i+j,\,\,c=i\] and \[(a\times b)\times c=\lambda \,a+\mu \,b\], then \[\lambda +\mu =\] [EAMCET 2003]
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
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question_answer20)
If the vectors a and b are mutually perpendicular, then \[\mathbf{a}\times \{\mathbf{a}\times \{\mathbf{a}\times (\mathbf{a}\times \mathbf{b})\}\}\] is equal to
A)
\[|\mathbf{a}{{|}^{2}}\mathbf{b}\] done
clear
B)
\[|\mathbf{a}{{|}^{3}}\mathbf{b}\] done
clear
C)
\[|\mathbf{a}{{|}^{4}}\mathbf{b}\] done
clear
D)
None of these done
clear
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question_answer21)
If a, b, c, d are coplanar vectors, then \[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=\] [MP PET 1998]
A)
\[|\,\mathbf{a}\,\times \,\mathbf{c}{{|}^{2}}\] done
clear
B)
\[|\mathbf{a}\times \mathbf{d}{{|}^{2}}\] done
clear
C)
\[|\mathbf{b}\times \mathbf{c}{{|}^{2}}\] done
clear
D)
0 done
clear
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question_answer22)
\[a\times [a\times (a\times \mathbf{b})]\] is equal to [AMU 2001]
A)
\[(a\times a\,)\,.\,(b\times a)\] done
clear
B)
\[a\,.\,(b\times a)-b\,.\,(a\times b)\] done
clear
C)
\[[a\,.\,(a\times b)]\,a\] done
clear
D)
\[(a\,.\,a)\,(b\times a)\] done
clear
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question_answer23)
Given the following simultaneous equations for vectors x and y \[\mathbf{x}+\mathbf{y}=\mathbf{a}\] .....(i) \[\mathbf{x}\times \mathbf{y}=\mathbf{b}\] .....(ii) \[\mathbf{x}\,.\,\mathbf{a}=1\] .....(iii) Then \[\mathbf{x}=.........,\,\,\,\mathbf{y}=.......\] [Roorkee 1994]
A)
\[\mathbf{a},\,\,\mathbf{a}-\mathbf{x}\] done
clear
B)
\[\mathbf{a}-\mathbf{b},\,\,\mathbf{b}\] done
clear
C)
\[\mathbf{b},\,\,\mathbf{a}-b\] done
clear
D)
None of these done
clear
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question_answer24)
\[(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})=\] [MP PET 1997]
A)
[b c a] a done
clear
B)
[c a b] b done
clear
C)
[a b c] c done
clear
D)
[a c b] b done
clear
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