JEE Main & Advanced Mathematics Vector Algebra Question Bank

Scaler or Dot product of two vectors and its application Total Questions - 100

1)

\[(\mathbf{a}\,.\,\mathbf{i})\,\mathbf{i}+(\mathbf{a}\,.\,\mathbf{j})\mathbf{j}+(\mathbf{a}\,.\,\mathbf{k})\,\mathbf{k}=\]                                                            [Karnataka CET 2004]

A)
            a

B)
            2 a

C)
            0

D)
            None of these
View Solution
2)

If \[\mathbf{r}\,.\,\mathbf{i}=\mathbf{r}\,.\,\mathbf{j}=\mathbf{r}\,.\,\mathbf{k}\] and \[|\mathbf{r}|\,\,=3,\] then \[\mathbf{r}=\]

A)
            \[\pm \,3\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]

B)
            \[\pm \,\frac{1}{3}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]

C)
            \[\pm \,\frac{1}{\sqrt{3}}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]

D)
            \[\pm \,\sqrt{3}\,(\mathbf{i}+\mathbf{j}+\mathbf{k})\]
View Solution
3)

If a, b, c are non-zero vectors such that \[\mathbf{a}\,\,.\,\,\mathbf{b}=\mathbf{a}\,\,.\,\,\mathbf{c},\] then which statement is true                                    [RPET 2001]

A)
            b = c

B)
            \[\mathbf{a}\,\bot \,(\mathbf{b}-\mathbf{c})\]

C)
            \[\mathbf{b}=\mathbf{c}\] or \[\mathbf{a}\,\bot \,(\mathbf{b}-\mathbf{c})\]

D)
            None of these
View Solution
4)

If a and b be unlike vectors, then a . b =

A)
            | a | | b |

B)
            ? | a | | b |

C)
            0

D)
            None of these
View Solution
5)

If a, b, c are unit vectors such that \[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0},\] then \[\mathbf{a}\,\,.\,\,\mathbf{b}+\mathbf{b}\,\,.\,\,\mathbf{c}+\mathbf{c}\,\,.\,\,\mathbf{a}=\] [MP PET 1988; Karnataka CET 2000; UPSEAT 2003, 04]

A)
            1

B)
            3

C)
            ? 3/2

D)
            3/2
View Solution
6)

If a, b, c are mutually perpendicular vectors of equal magnitudes, then the angle between the vectors a and \[\mathbf{a}+\mathbf{b}+\mathbf{c}\] is

A)
            \[\frac{\pi }{3}\]

B)
            \[\frac{\pi }{6}\]

C)
            \[{{\cos }^{-1}}\frac{1}{\sqrt{3}}\]

D)
            \[\frac{\pi }{2}\]
View Solution
7)

If a, b, c are mutually perpendicular unit vectors, then \[|\mathbf{a}+\mathbf{b}+\mathbf{c}|\,\,=\]         [Karnataka CET 2002, 05; J & K 2005]

A)
            \[\sqrt{3}\]

B)
            3

C)
            1

D)
            0
View Solution
8)

If \[|\mathbf{a}|+|\mathbf{b}|\,=\,|\mathbf{c}|\] and \[\mathbf{a}+\mathbf{b}=\mathbf{c},\] then the angle between a and b is

A)
            \[\frac{\pi }{2}\]

B)
            \[\pi \]

C)
            0

D)
            None of these
View Solution
9)

If a has magnitude 5 and points north-east and vector b has magnitude 5 and points north-west, then \[|\,\,\mathbf{a}-\mathbf{b}\,\,|\,=\]                                     [MNR 1984]

A)
            25

B)
            5

C)
            \[7\sqrt{3}\]

D)
            \[5\sqrt{2}\]
View Solution
10)

If q be the angle between the unit vectors a and b, then \[\cos \frac{\theta }{2}=\] [MP PET 1998; Pb. CET 2002]

A)
            \[\frac{1}{2}\,|\mathbf{a}-\mathbf{b}|\]

B)
            \[\frac{1}{2}\,|\mathbf{a}+\mathbf{b}|\]

C)
            \[\frac{|\mathbf{a}-\mathbf{b}|}{|\mathbf{a}+\mathbf{b}|}\]

D)
            \[\frac{|\mathbf{a}+\mathbf{b}|}{|\mathbf{a}-\mathbf{b}|}\]
View Solution
11)

If \[|\mathbf{a}|\,=3,\,\,|\mathbf{b}|\,=4,\,\,|\mathbf{c}|\,=\,\,5\] and \[\mathbf{a}+\mathbf{b}+\mathbf{c}=0,\] then the angle between a and b is   [MP PET 1989; Bihar CEE 1994]

A)
            0

B)
            \[\frac{\pi }{6}\]

C)
            \[\frac{\pi }{3}\]

D)
            \[\frac{\pi }{2}\]
View Solution
12)

If \[|\mathbf{a}+\mathbf{b}|\,\,>\,\,|\mathbf{a}-\mathbf{b}|,\] then the angle between a and b is

A)
            Acute

B)
            Obtuse

C)
            \[\frac{\pi }{2}\]

D)
            \[\pi \]
View Solution
13)

If a, b, c are three vectors such that \[\mathbf{a}=\mathbf{b}+\mathbf{c}\] and the angle between b and c is \[\pi /2,\] then             [EAMCET 2003]

A)
            \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}\]

B)
            \[{{b}^{2}}={{c}^{2}}+{{a}^{2}}\]

C)
            \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\]

D)
            \[2{{a}^{2}}-{{b}^{2}}={{c}^{2}}\] (Note : Here \[a=\,\,|\mathbf{a}|,\,\,b=\,|\,\mathbf{b}|,\,\,c=\,|\mathbf{c}|)\]
View Solution
14)

If the angle between the vectors a and b be q and \[\mathbf{a}\,.\,\mathbf{b}=\cos \theta ,\] then the true statement is

A)
            a and b are equal vectors

B)
            a and b are like vectors

C)
            a and b are unlike vectors

D)
            a and b are unit vectors
View Solution
15)

If the vector \[\mathbf{i}+\mathbf{j}+\mathbf{k}\] makes angles \[\alpha ,\,\beta ,\,\gamma \]with vectors \[\mathbf{i},\,\mathbf{j},\mathbf{k}\]respectively, then

A)
            \[\alpha =\beta \ne \gamma \]

B)
            \[\alpha =\gamma \ne \beta \]

C)
            \[\beta =\gamma \ne \alpha \]

D)
            \[\alpha =\beta =\gamma \]
View Solution
16)

\[{{(\mathbf{r}\,.\,\mathbf{i})}^{2}}+{{(\mathbf{r}\,.\,\mathbf{j})}^{2}}+{{(\mathbf{r}\,.\,\mathbf{k})}^{2}}=\]

A)
            \[3{{r}^{2}}\]

B)
            \[{{r}^{2}}\]

C)
            0

D)
            None of these
View Solution
17)

The value of b such that scalar product of the vectors \[(\mathbf{i}+\mathbf{j}+\mathbf{k})\] with the unit vector parallel to the sum of the vectors \[(2\mathbf{i}+4\mathbf{j}-5\mathbf{k})\] and \[(b\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] is 1, is [MNR 1992; Roorkee 1985, 95; Kurukshetra CEE 1998; UPSEAT 2000]

A)
            ? 2

B)
            ? 1

C)
            0

D)
            1
View Solution
18)

If a unit vector lies in yz?plane and makes angles of \[{{30}^{o}}\] and \[{{60}^{o}}\] with the positive y-axis and z-axis respectively, then its components along the co-ordinate axes will be

A)
            \[\frac{\sqrt{3}}{2},\,\,\frac{1}{2},\,0\]

B)
            \[0,\,\,\frac{\sqrt{3}}{2},\,\,\frac{1}{2}\]

C)
            \[\frac{\sqrt{3}}{2},\,\,0,\,\,\frac{1}{2}\]

D)
            \[0,\,\,\frac{1}{2},\,\frac{\sqrt{3}}{2}\]
View Solution
19)

If \[\overrightarrow{{{F}_{1}}}=\mathbf{i}-\mathbf{j}+\mathbf{k},\] \[\overrightarrow{{{F}_{2}}}=-\mathbf{i}+2\mathbf{j}-\mathbf{k},\] \[\overrightarrow{{{F}_{3}}}=\mathbf{j}-\mathbf{k},\] \[\vec{A}=4\mathbf{i}-3\mathbf{j}-2\mathbf{k}\] and \[\vec{B}=6\mathbf{i}+\mathbf{j}-3\mathbf{k},\] then the scalar product of \[\overrightarrow{{{F}_{1}}}+\overrightarrow{{{F}_{2}}}+\overrightarrow{{{F}_{3}}}\]and \[\overrightarrow{AB}\] will be            [Roorkee 1980]

A)
            3

B)
            6

C)
            9

D)
            12
View Solution
20)

If the moduli of a and b are equal and angle between them is \[{{120}^{o}}\] and \[\mathbf{a}\,.\,\mathbf{b}=-\,8,\] then | a | is equal to [RPET 1986]

A)
            ? 5

B)
            ? 4

C)
            4

D)
            5
View Solution
21)

If \[|\mathbf{a}|\,\,=3,\,\,\,|\mathbf{b}\,\,|\,\,=4\] and the angle between a and b be \[{{120}^{o}}\], then \[|4\mathbf{a}+3\mathbf{b}|\,\,=\]

A)
            25

B)
            12

C)
            13

D)
            7
View Solution
22)

A vector whose modulus is \[\sqrt{51}\] and makes the same angle with \[\mathbf{a}=\frac{\mathbf{i}-2\mathbf{j}+2\mathbf{k}}{3},\,\,\mathbf{b}=\frac{-\,4\mathbf{i}-3\mathbf{k}}{5}\] and \[\mathbf{c}=\mathbf{j},\] will be                                     [Roorkee 1987]

A)
            \[5\mathbf{i}+5\mathbf{j}+\mathbf{k}\]

B)
            \[5\mathbf{i}+\mathbf{j}-5\mathbf{k}\]

C)
            \[5\mathbf{i}+\mathbf{j}+5\mathbf{k}\]

D)
            \[\pm \,(5\mathbf{i}-\mathbf{j}-5\mathbf{k})\]
View Solution
23)

If a, b, c are coplanar vectors, then     [IIT 1989]

A)
            \[\left| \,\begin{matrix}    \mathbf{a} & \mathbf{b} & \mathbf{c} \\    \mathbf{b} & \mathbf{c} & \mathbf{a} \\    \mathbf{c} & \mathbf{a} & \mathbf{b} \\ \end{matrix}\, \right|=\mathbf{0}\]

B)
            \[\left| \,\begin{matrix}    \mathbf{a} & \mathbf{b} & \mathbf{c} \\    \mathbf{a}\,.\,\mathbf{a} & \mathbf{a}\,.\,\mathbf{b} & \mathbf{a}\,.\,\mathbf{c} \\    \mathbf{b}\,.\,\mathbf{a} & \mathbf{b}\,.\,\mathbf{b} & \mathbf{b}\,.\,\mathbf{c} \\ \end{matrix}\, \right|=\mathbf{0}\]

C)
            \[\left| \,\begin{matrix}    \mathbf{a} & \mathbf{b} & \mathbf{c} \\    \mathbf{c}\,.\,\mathbf{a} & \mathbf{c}\,.\,\mathbf{b} & \mathbf{c}\,.\,\mathbf{c} \\    \mathbf{b}\,.\,\mathbf{a} & \mathbf{b}\,.\,\mathbf{c} & \mathbf{b}\,.\,\mathbf{b} \\ \end{matrix}\, \right|=\mathbf{0}\]

D)
            \[\left| \,\begin{matrix}    \mathbf{a} & \mathbf{b} & \mathbf{c} \\    \mathbf{a}\,.\,\mathbf{b} & \mathbf{a}\,.\,\mathbf{a} & \mathbf{a}\,.\,\mathbf{c} \\    \mathbf{c}\,.\,\mathbf{a} & \mathbf{c}\,.\,\mathbf{c} & \mathbf{c}\,.\,\mathbf{b} \\ \end{matrix}\, \right|=\mathbf{0}\]
View Solution
24)

If \[\vec{\lambda }\] is a unit vector perpendicular to plane of vector a and b and angle between them is q, then a . b will be                                                 [RPET 1985]

A)
            \[|\mathbf{a}|\,\,|\mathbf{b}|\,\sin \theta \,\vec{\lambda }\]

B)
            \[|\mathbf{a}|\,\,|\mathbf{b}|\,\cos \,\,\theta \,\vec{\lambda }\]

C)
            \[|\mathbf{a}|\,\,|\mathbf{b}|\,\cos \,\,\theta \,\]

D)
            \[|\mathbf{a}|\,\,|\mathbf{b}|\,\sin \theta \,\]
View Solution
25)

If \[\mathbf{p}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[\mathbf{q}=3\mathbf{i}+\mathbf{j}+2\mathbf{k},\] then a vector along r which is linear combination of p and q and also perpendicular to q is                           [MNR 1986]

A)
            \[\mathbf{i}+5\mathbf{j}-4\mathbf{k}\]

B)
            \[\mathbf{i}-5\mathbf{j}+4\mathbf{k}\]

C)
            \[-\frac{1}{2}\,(\mathbf{i}+5\mathbf{j}-4\mathbf{k})\]

D)
            None of these
View Solution
26)

If \[\mathbf{d}=\lambda \,(\mathbf{a}\times \mathbf{b})+\mu \,(\mathbf{b}\times \mathbf{c})+\nu \,(\mathbf{c}\times \mathbf{a})\]and \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]=\frac{1}{8},\] then \[\lambda +\mu +\nu \] is equal to

A)
            \[8\mathbf{d}\,.\,(\mathbf{a}+\mathbf{b}+\mathbf{c})\]

B)
            \[8\mathbf{d}\,\times \,(\mathbf{a}+\mathbf{b}+\mathbf{c})\]

C)
            \[\frac{\mathbf{d}\,}{8}.\,(\mathbf{a}+\mathbf{b}+\mathbf{c})\]

D)
            \[\frac{\mathbf{d}\,}{8}\times \,(\mathbf{a}+\mathbf{b}+\mathbf{c})\]
View Solution
27)

The horizontal force and the force inclined at an angle \[{{60}^{o}}\] with the vertical, whose resultant is in vertical direction of       P kg, are                                 [IIT 1983]

A)
            \[P,2P\]

B)
            \[P,\,\,P\sqrt{3}\]

C)
            \[2P,\,\,P\sqrt{3}\]

D)
            None of these
View Solution
28)

If a and b are mutually perpendicular vectors, then \[{{(\mathbf{a}+\mathbf{b})}^{2}}=\]         [MP PET 1994; Pb. CET 2002]

A)
            \[\mathbf{a}+\mathbf{b}\]

B)
            \[\mathbf{a}-\mathbf{b}\]

C)
            \[{{a}^{2}}-{{b}^{2}}\]

D)
            \[{{(\mathbf{a}-\mathbf{b})}^{2}}\]
View Solution
29)

\[\mathbf{a}\,.\,\mathbf{b}=0,\] then [RPET 1995]

A)
            \[\mathbf{a}\,\bot \,\mathbf{b}\]

B)
            a || b

C)
            Angle between a and b is \[{{60}^{o}}\]

D)
            None of these
View Solution
30)

If \[|\mathbf{a}|\,\,=3,\,\,\,|\mathbf{b}|\,\,=1,\,\,|\mathbf{c}|\,\,=4\] and \[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0},\] then \[\mathbf{a}\,.\,\mathbf{b}+\mathbf{b}\,.\,\mathbf{c}+\mathbf{c}\,.\,\mathbf{a}=\] [MP PET 1995; RPET 2000]

A)
            ? 13

B)
            ? 10

C)
            13

D)
            10
View Solution
31)

If ABCDEF is regular hexagon, the length of whose side is a, then \[\overrightarrow{AB}\,\,.\,\overrightarrow{AF}+\frac{1}{2}\,{{\overrightarrow{BC}}^{2}}=\]

A)
            a

B)
            \[{{a}^{2}}\]

C)
            \[2\,{{a}^{2}}\]

D)
            0
View Solution
32)

If in a right angled triangle ABC, the hypotenuse \[AB=p,\] then \[\overrightarrow{AB}\,\,.\,\,\overrightarrow{AC}+\overrightarrow{BC}\,.\,\,\overrightarrow{BA}+\overrightarrow{CA}\,\,.\,\,\overrightarrow{CB}\] is equal to

A)
            \[2{{p}^{2}}\]

B)
            \[\frac{{{p}^{2}}}{2}\]

C)
            \[{{p}^{2}}\]

D)
            None of these
View Solution
33)

A, B, C, D are any four points, then \[\overrightarrow{AB}\,\,.\,\,\overrightarrow{CD}\,\,+\,\overrightarrow{\,BC}\,\,.\,\,\overrightarrow{AD}\,\,+\overrightarrow{CA}\,\,.\,\,\overrightarrow{BD}\,\,=\]            [MNR 1986]

A)
            \[2\,\,\overrightarrow{AB}\,\,.\,\,\overrightarrow{BC}\,\,.\,\,\overrightarrow{CD}\]

B)
            \[\overrightarrow{AB}\,\,+\,\,\overrightarrow{BC}\,\,+\,\,\overrightarrow{CD}\]

C)
            \[5\sqrt{3}\]

D)
            0
View Solution
34)

The vector a coplanar with the vectors i and j, perpendicular to the vector \[\mathbf{b}=4\mathbf{i}-3\mathbf{j}+5\mathbf{k}\] such that \[|\mathbf{a}|\,=\,|\,\mathbf{b}|\] is

A)
            \[\sqrt{2}\,(3\mathbf{i}+4\mathbf{j})\] or \[-\sqrt{2}\,(3\mathbf{i}+4\mathbf{j})\]

B)
            \[\sqrt{2}\,(4\mathbf{i}+3\mathbf{j})\] or \[-\sqrt{2}\,(4\mathbf{i}+3\mathbf{j})\]

C)
            \[\sqrt{3}\,(4\mathbf{i}+5\mathbf{j})\] or \[-\sqrt{3}\,(4\mathbf{i}+5\mathbf{j})\]

D)
            \[\sqrt{3}\,(5\mathbf{i}+4\mathbf{j})\] or \[-\sqrt{3}\,(5\mathbf{i}+4\mathbf{j})\]
View Solution
35)

If a is any vector in space, then [MP PET 1997]

A)
\[\mathbf{a}=(\mathbf{a}\,.\,\mathbf{i})\,\mathbf{i}+\,(\mathbf{a}\,.\,\mathbf{j})\,\mathbf{j}+\,(\mathbf{a}\,.\,\mathbf{k})\,\mathbf{k}\]

B)
\[\mathbf{a}=(\mathbf{a}\,\times \,\mathbf{i})+\,(\mathbf{a}\,\times \,\mathbf{j})\,+\,(\mathbf{a}\,\times \,\mathbf{k})\,\]

C)
\[\mathbf{a}=\mathbf{j}\,(\mathbf{a}\,.\,\mathbf{i})\,+\mathbf{k}\,(\mathbf{a}\,.\,\mathbf{j})\,+\,\mathbf{i}\,(\mathbf{a}\,.\,\mathbf{k})\,\]

D)
\[\mathbf{a}=(\mathbf{a}\,\times \,\mathbf{i})\times \mathbf{i}+\,(\mathbf{a}\,\times \,\mathbf{j})\times \mathbf{j}\,+\,(\mathbf{a}\,\times \,\mathbf{k})\times \mathbf{k}\,\]
View Solution
36)

If vectors \[\mathbf{a},\,b,\,\mathbf{c}\] satisfy the condition \[|\mathbf{a}-\mathbf{c}|=|\mathbf{b}-\mathbf{c}|\], then \[(\mathbf{b}-\mathbf{a})\,.\,\left( \mathbf{c}-\frac{\mathbf{a}+\mathbf{b}}{\mathbf{2}} \right)\]is equal to [AMU 1999]

A)
            0

B)
            ?1

C)
            1

D)
            2
View Solution
37)

(a .b) c and (a.c) b are [RPET 2000]

A)
            Two like vectors

B)
            Two equal vectors

C)
            Two vectors in direction of a

D)
            None of these
View Solution
38)

If \[a=(1,\,-1,\,2),\ b=(-2,\,3,\,5)\], \[\mathbf{c}=(2\,,\,-2,\,4)\] and i is the unit vector in the x-direction, then \[(a-2b+3c)\,.\,i=\] [Karnataka CET 2001]

A)
            11

B)
            15

C)
            18

D)
            36
View Solution
39)

For any three non-zero vectors \[{{r}_{1}},\,{{r}_{2}}\] and \[{{r}_{3}}\], \[\left| \,\begin{matrix}    {{r}_{1}}\,.\,{{r}_{1}} & {{r}_{1}}\,.\,{{r}_{2}} & {{r}_{1}}\,.\,{{r}_{3}} \\    {{r}_{2}}\,.\,{{r}_{1}} & {{r}_{2}}\,.\,{{r}_{2}} & {{r}_{2}}\,.\,{{r}_{3}} \\    {{r}_{3}}\,.\,{{r}_{1}} & {{r}_{3}}\,.\,{{r}_{2}} & {{r}_{3}}\,.\,{{r}_{3}} \\ \end{matrix} \right|=0\]. Then which of the following is false                                    [AMU 2000]

A)
            All the three vectors are parallel to one and the same plane

B)
            All the three vectors are linearly dependent

C)
            This system of equation has a non-trivial solution

D)
            All the three vectors are perpendicular to each other
View Solution
40)

Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a + b + c = 0, then the values of                 a.b + b.c + c.a is           [IIT 1995; DCE 2001; AIEEE 2002; UPSEAT 2002; Kerala (Engg.) 2005]

A)
            47

B)
            25

C)
            50

D)
            ? 25
View Solution
41)

If a and b are adjacent sides of a rhombus, then        [RPET 2001]

A)
            a.b = 0

B)
            a × b = 0

C)
            a.a = b.b

D)
            None of these
View Solution
42)

If x and y are two unit vectors and \[\pi \] is the angle between them, then \[\frac{1}{2}|x-y|\] is equal to             [UPSEAT 2001]

A)
            0

B)
            \[\pi /2\]

C)
            1

D)
            \[\pi /4\]
View Solution
43)

If \[a\,.\,i=a\,.\,(i+j)=a\,.\,(i+j+k)\], then a =           [EAMCET 2002]

A)
          i

B)
            k

C)
            j

D)
            i + j + k
View Solution
44)

If i, j, k are unit vectors, then                          [MP PET 2001]

A)
            i . j \[=\]1

B)
            i . i \[=\]1

C)
            \[\mathbf{i}\times \mathbf{j}=1\]

D)
            \[\mathbf{i}\times (\mathbf{j}\times \mathbf{k})=1\]
View Solution
45)

If \[|a|\,=\,|\mathbf{b}|,\] then \[(a+b)\,.\,(a-b)\] is             [MP PET 2002]

A)
            Positive

B)
            Negative

C)
            Zero

D)
            None of these
View Solution
46)

\[a,\,b,\,c\] are three vectors, such that \[a+b+c=0\], \[|a|\,=1,\,|b|\,=2,\,|c|\,=3\], then \[a.b+b.c+c.a\] is equal to [AIEEE 2003]

A)
            0

B)
            ? 7

C)
            7

D)
            1
View Solution
47)

A unit vector which is coplanar to vector \[\mathbf{i}+\mathbf{j}+2k\] and \[\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and perpendicular to \[\mathbf{i}+\mathbf{j}+\mathbf{k},\] is [IIT 1992; Kurukshetra CEE 2002]

A)
            \[\frac{\mathbf{i}-\mathbf{j}}{\sqrt{2}}\]

B)
            \[\pm \,\left( \frac{\mathbf{j}-\mathbf{k}}{\sqrt{2}} \right)\]

C)
            \[\frac{\mathbf{k}-\mathbf{i}}{\sqrt{2}}\]

D)
            \[\frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\]
View Solution
48)

If \[|\mathbf{a}|\,=3,\,\,|\mathbf{b}|\,=4\] then a value of l for which \[\mathbf{a}+\lambda \mathbf{b}\] is perpendicular to \[\mathbf{a}-\lambda \mathbf{b}\] is                                   [Karnataka CET 2004]

A)
            \[\mathbf{a}=2\,\mathbf{i}+2\,\mathbf{j}+3\,\mathbf{k},\,\,\mathbf{b}=-\mathbf{i}+2\,\mathbf{j}+\mathbf{k}\]

B)
            \[\frac{3}{4}\]

C)
            \[\frac{3}{2}\]

D)
            \[\frac{4}{3}\]
View Solution
49)

\[\mathbf{a},\,\mathbf{b}\] and c are three vectors with magnitude \[|\mathbf{a}|\,=4,\] \[|\mathbf{b}|\,=4,\] \[|\mathbf{c}|\,=2\] and such that \[\mathbf{a}\] is perpendicular to \[(\mathbf{b}+\mathbf{c}),\,\mathbf{b}\] is perpendicular to \[(\mathbf{c}+\mathbf{a})\] and \[\mathbf{c}\] is perpendicular to \[(\mathbf{a}+\mathbf{b}).\] It follows that \[|\mathbf{a}+\mathbf{b}+\mathbf{c}|\] is equal to           [UPSEAT 2004]

A)
            9

B)
            6

C)
            5

D)
            4
View Solution
50)

The angle between the vectors \[3\,\mathbf{i}+\mathbf{j}+2\,\mathbf{k}\] and \[2\,\mathbf{i}-2\,\mathbf{j}+4\,\mathbf{k}\] is                                [MP PET 1990]

A)
            \[{{\cos }^{-1}}\frac{2}{\sqrt{7}}\]

B)
            \[{{\sin }^{-1}}\frac{2}{\sqrt{7}}\]

C)
            \[{{\cos }^{-1}}\frac{2}{\sqrt{5}}\]

D)
            \[{{\sin }^{-1}}\frac{2}{\sqrt{5}}\]
View Solution
51)

If the position vectors of the points A, B, C, D be \[\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,2\,\mathbf{i}+5\,\mathbf{j},\,\,3\,\mathbf{i}+2\,\mathbf{j}-3\mathbf{k}\]and \[\mathbf{i}-6\,\mathbf{j}-\mathbf{k},\] then the angle between the vectors \[\overrightarrow{AB}\] and \[\overrightarrow{CD}\] is

A)
            \[\frac{\pi }{4}\]

B)
            \[\frac{\pi }{3}\]

C)
            \[\frac{\pi }{2}\]

D)
            \[\pi \]
View Solution
52)

If q be the angle between the unit vectors a and b, then \[\mathbf{a}-\sqrt{2}\,\mathbf{b}\] will be a unit vector if \[\theta =\]

A)
            \[\frac{\pi }{6}\]

B)
            \[\frac{\pi }{4}\]

C)
            \[p\mathbf{i}+q\mathbf{j}+r\mathbf{k}\]

D)
            \[\frac{2\pi }{3}\]
View Solution
53)

If the angle between a and b be \[{{30}^{o}}\], then the angle between 3 a and ? 4 b will be

A)
            \[{{150}^{o}}\]

B)
            \[{{90}^{o}}\]

C)
            \[{{120}^{o}}\]

D)
            \[{{30}^{o}}\]
View Solution
54)

The angle between the vectors \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+2\mathbf{j}+\mathbf{k}\] is                                     [BIT Ranchi 1991]

A)
            \[{{\cos }^{-1}}\left( \frac{1}{\sqrt{15}} \right)\]

B)
            \[{{\cos }^{-1}}\left( \frac{4}{\sqrt{15}} \right)\]

C)
            \[{{\cos }^{-1}}\left( \frac{4}{15} \right)\]

D)
            \[\frac{\pi }{2}\]
View Solution
55)

The position vector of vertices of a triangle ABC are \[4\mathbf{i}-2\mathbf{j},\,\mathbf{i}+4\mathbf{j}-3\mathbf{k}\] and \[-\mathbf{i}+5\mathbf{j}+\mathbf{k}\] respectively, then \[\angle ABC=\] [RPET 1988, 97]

A)
            \[\pi /6\]

B)
            \[\pi /4\]

C)
            \[\pi /3\]

D)
            \[\pi /2\]
View Solution
56)

The value of x for which the angle between the vectors \[\mathbf{a}=x\mathbf{i}-3\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=2x\mathbf{i}+x\mathbf{j}-\mathbf{k}\] is acute and the angle between the vectors b and the axis of ordinate is obtuse, are

A)
            1, 2

B)
            ? 2, ? 3

C)
            x > 0

D)
            None of these
View Solution
57)

If a and b are unit vectors and \[\mathbf{a}-\mathbf{b}\] is also a unit vector, then the angle between a and b is [RPET 1991; MP PET 1995; Pb. CET 2001]

A)
            \[\frac{\pi }{4}\]

B)
            \[\frac{\pi }{3}\]

C)
            \[\frac{\pi }{2}\]

D)
            \[\frac{2\pi }{3}\]
View Solution
58)

If q be the angle between two vectors a and b, then                     \[\mathbf{a}.\mathbf{b}\] \[\ge 0\] if                                 [MP PET 1995]

A)
            \[0\le \theta \le \pi \]

B)
            \[\frac{\pi }{2}\le \theta \le \pi \]

C)
            \[0\le \theta \le \frac{\pi }{2}\]

D)
            None of these
View Solution
59)

If \[\mathbf{a}=\mathbf{i}+2\mathbf{j}-3\mathbf{k}\] and \[\mathbf{b}=3\mathbf{i}-\mathbf{j}+2\mathbf{k},\] then the angle between the vectors \[\mathbf{a}+\mathbf{b}\] and \[\mathbf{a}-\mathbf{b}\] is            [Karnataka CET 1994; Orissa JEE 2005]

A)
            \[{{30}^{o}}\]

B)
            \[{{60}^{o}}\]

C)
            \[{{90}^{o}}\]

D)
            x
View Solution
60)

The value of x for which the angle between the vectors \[\mathbf{a}=-\,3\mathbf{i}+x\mathbf{j}+\mathbf{k}\] and \[\mathbf{b}=x\mathbf{i}+2x\mathbf{j}+\mathbf{k}\] is acute and the angle between b and x-axis lies between \[\pi /2\] and \[\pi \]satisfy                                     [Kurukshetra CEE 1996]

A)
            \[x>0\]

B)
            \[x<0\]

C)
            \[x>1\] only

D)
            \[x<-1\] only
View Solution
61)

The angle between the vectors \[(2\mathbf{i}+6\mathbf{j}+3\mathbf{k})\] and \[(12\mathbf{i}-4\mathbf{j}+3\mathbf{k})\] is                                   [MP PET 1996]

A)
            \[{{\cos }^{-1}}\left( \frac{1}{10} \right)\]

B)
            \[{{\cos }^{-1}}\left( \frac{9}{11} \right)\]

C)
            \[{{\cos }^{-1}}\left( \frac{9}{91} \right)\]

D)
            \[{{\cos }^{-1}}\left( \frac{1}{9} \right)\]
View Solution
62)

If the angle between two vectors \[\mathbf{i}+\mathbf{k}\] and \[\mathbf{i}-\mathbf{j}+a\mathbf{k}\] is \[\pi /3,\] then the value of \[a=\]                            [MP PET 1997]

A)
            2

B)
            4

C)
            ? 2

D)
            0
View Solution
63)

If three vectors a, b, c satisfy \[\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\] and \[|\mathbf{a}|\,\,=\,\,3,\,\] \[|\mathbf{b}|\,=5,\] \[|\mathbf{c}|\,\,=7,\] then the angle between a and b is [Kurukshetra CEE 1998; UPSEAT 2001; AIEEE 2002; MP PET 2002]

A)
            \[{{30}^{o}}\]

B)
            \[{{45}^{o}}\]

C)
            \[{{60}^{o}}\]

D)
            \[{{90}^{\text{o}}}\]
View Solution
64)

If a, b and c are unit vectors such that \[\mathbf{a}+\mathbf{b}-\mathbf{c}=0,\] then the angle between a and b is [Roorkee Qualifying 1998; MP PET 1999; UPSEAT 2000; RPET 2002]

A)
            \[\pi /6\]

B)
            \[\pi /3\]

C)
            \[\pi /2\]

D)
            \[2\pi /3\]
View Solution
65)

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is [Kurukshetra CEE 1996; RPET 1996]

A)
            \[\sqrt{2}\]

B)
            \[\sqrt{3}\]

C)
            \[\frac{1}{\sqrt{3}}\]

D)
            1
View Solution
66)

The angle between the vector \[2i+3j+k\] and \[2i-j-k\] is       [MNR 1990; UPSEAT 2000]

A)
            \[\pi /2\]

B)
            \[\pi /4\]

C)
            \[\pi /3\]

D)
            0
View Solution
67)

If \[\theta \] be the angle between the vectors \[12+m-2=0\] and \[\mathbf{b}=6\mathbf{i}-3\mathbf{j}+2\mathbf{k}\], then [MP PET 2001, 03]

A)
            \[\cos \theta =\frac{4}{21}\]

B)
            \[\cos \theta =\frac{3}{19}\]

C)
            \[\cos \theta =\frac{2}{19}\]

D)
            \[\cos \theta =\frac{5}{21}\]
View Solution
68)

If a and b are two unit vectors such that \[\mathbf{a}+2\,\mathbf{b}\] and \[5a-4b\] are perpendicular to each other, then the angle between a and b is [IIT Screening 2002]

A)
            \[{{45}^{o}}\]

B)
            \[{{60}^{o}}\]

C)
            \[{{\cos }^{-1}}\left( \frac{1}{3} \right)\]

D)
            \[{{\cos }^{-1}}\left( \frac{2}{7} \right)\]
View Solution
69)

Let a and b be two unit vectors inclined at an angle \[\theta \], then \[\sin \,(\theta /2)\] is equal to [BIT Ranchi 1991; Karnataka CET 2000, 01; UPSEAT 2002]

A)
            \[\frac{1}{2}|a-b|\]

B)
            \[\frac{1}{2}|a+b|\]

C)
            \[|a-b|\]

D)
            \[|a+b|\]
View Solution
70)

The angle between the vectors a + b and a ? b, when \[\mathbf{a}=(1,\,1,\,4)\] and \[b=(1,\,-1,\,4)\] is [Karnataka CET 2003]

A)
            \[{{90}^{o}}\]

B)
            \[{{45}^{o}}\]

C)
            \[{{30}^{o}}\]

D)
            \[{{15}^{o}}\]
View Solution
71)

A vector of length 3 perpendicular to each of the vectors \[3\,\mathbf{i}+\mathbf{j}-4\,\mathbf{k}\] and \[6\,\mathbf{i}+5\,\mathbf{j}-2\,\mathbf{k}\] is

A)
            \[2\,\mathbf{i}-2\,\mathbf{j}+\mathbf{k}\]

B)
            \[-\,2\,\mathbf{i}+2\,\mathbf{j}+\mathbf{k}\]

C)
            \[2\,\mathbf{i}+2\,\mathbf{j}-\mathbf{k}\]

D)
            None of these
View Solution
72)

If \[\mathbf{a}\ne \mathbf{0},\,\,\mathbf{b}\ne \mathbf{0}\] and \[|\mathbf{a}+\mathbf{b}|\,=\,|\mathbf{a}-\mathbf{b}|,\] then the vectors a and b are [Roorkee 1986; MNR 1988; IIT Screening 1989; MP PET 1990, 97; RPET 1984, 90, 96, 99; KCET 1999]

A)
            Parallel to each other

B)
            Perpendicular to each other

C)
            Inclined at an angle of \[{{60}^{o}}\]

D)
            Neither perpendicular nor parallel
View Solution
73)

The vector \[2\,\mathbf{i}+a\,\mathbf{j}+\mathbf{k}\] is perpendicular to the vector \[2\,\mathbf{i}-\mathbf{j}-k,\] if \[a=\]                             [MP PET 1987]

A)
            5

B)
            ? 5

C)
            ? 3

D)
            3
View Solution
74)

If \[\mathbf{a}=2\,\mathbf{i}+2\,\mathbf{j}+3\,\mathbf{k},\,\,\mathbf{b}=-\mathbf{i}+2\,\mathbf{j}+\mathbf{k}\] and \[c=3\,\mathbf{i}+\mathbf{j},\] then \[\mathbf{a}+t\,\mathbf{b}\] is perpendicular to c if \[t=\] [MNR 1979; MP PET 2002]

A)
            2

B)
            4

C)
            6

D)
            8
View Solution
75)

The vector \[2\,\mathbf{i}+\mathbf{j}-\mathbf{k}\] is perpendicular to \[\mathbf{i}-4\mathbf{j}+\lambda \mathbf{k},\] if \[\lambda =\]              [MNR 1983; MP PET 1988]

A)
            0

B)
            ? 1

C)
            ? 2

D)
            ? 3
View Solution
76)

The vectors \[2\,\mathbf{i}+3\,\mathbf{j}-4\,\mathbf{k}\] and \[a\,\mathbf{i}+b\,\mathbf{j}+c\,\mathbf{k}\] are perpendicular, when [MNR 1982; MP PET 1988; MP PET 2002]

A)
            \[a=2,\,\,b=3,\,\,c=-4\]

B)
            \[a=4,\,\,b=4,\,\,c=5\]

C)
            \[a=4,\,\,b=4,\,\,c=-\,5\]

D)
            None of these
View Solution
77)

A unit vector in the \[xy-\]plane which is perpendicular to \[4\mathbf{i}-3\mathbf{j}+\mathbf{k}\] is                                  [RPET 1991]

A)
            \[\frac{\mathbf{i}+\mathbf{j}}{\sqrt{2}}\]

B)
            \[\frac{1}{5}(3\mathbf{i}+4\mathbf{j})\]

C)
            \[\frac{1}{5}\,(3\mathbf{i}-4\mathbf{j})\]

D)
            None of these
View Solution
78)

If \[l\,\mathbf{a}+m\,\mathbf{b}+n\,\mathbf{c}=\mathbf{0},\] where \[l,\,m,\,\,n\] are scalars and a, b, c are mutually perpendicular vectors, then

A)
            \[l=m=n=1\]

B)
            \[l+m+n=1\]

C)
            \[l=m=n=0\]

D)
            \[l\ne 0,\,\,m\ne 0,\,\,n\ne 0\]
View Solution
79)

The unit normal vector to the line joining \[\mathbf{i}-\mathbf{j}\] and \[2\,\mathbf{i}+3\,\mathbf{j}\] and pointing towards the origin is [MP PET 1989]

A)
            \[\frac{4\,\mathbf{i}-\mathbf{j}}{\sqrt{17}}\]

B)
            \[\frac{-4\,\mathbf{i}+\mathbf{j}}{\sqrt{17}}\]

C)
            \[\frac{2\,\mathbf{i}-3\,\mathbf{j}}{\sqrt{13}}\]

D)
            \[\frac{-\,2\,\mathbf{i}+3\,\mathbf{j}}{\sqrt{13}}\]
View Solution
80)

If the vectors \[a\,\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[3\mathbf{i}+6\mathbf{j}-5\mathbf{k}\] are perpendicular to each other, then a is given by [MP PET 1993]

A)
            9

B)
            16

C)
            25

D)
            36
View Solution
81)

The value of \[\lambda \] for which the vectors \[2\lambda \mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[2\mathbf{j}+\mathbf{k}\] are perpendicular, is                               [MP PET 1992]

A)
            None

B)
            ? 1

C)
            1

D)
            Any value
View Solution
82)

If the vectors \[a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\] and \[p\mathbf{i}+q\mathbf{j}+r\mathbf{k}\] are perpendicular, then [RPET 1989]

A)
            \[(a+b+c)\,(p+q+r)=0\]

B)
            \[(a+b+c)\,(p+q+r)=1\]

C)
            \[ap+bq+cr=0\]

D)
            \[ap+bq+cr=1\]
View Solution
83)

If \[\mathbf{a}=2\mathbf{i}+4\mathbf{j}+2\mathbf{k}\] and \[\mathbf{b}=8\mathbf{i}-3\mathbf{j}+\lambda \mathbf{k}\] and \[\mathbf{a}\,\bot \,\mathbf{b},\] then value of \[\lambda \] will be [RPET 1995]

A)
            2

B)
            ? 1

C)
            ? 2

D)
            1
View Solution
84)

The vector \[\frac{1}{3}\,(2\mathbf{i}-2\mathbf{j}+\mathbf{k})\] is                                   [IIT Screening 1994]

A)
            A unit vector

B)
            Makes an angle \[\frac{\pi }{3}\] with the vector \[2i-4\mathbf{j}+3\mathbf{k}\]

C)
            Parallel to the vector \[-\mathbf{i}+\mathbf{j}-\frac{1}{2}\mathbf{k}\]

D)
          Perpendicular to the vector \[3\mathbf{i}+2\mathbf{j}-2\mathbf{k}\]
View Solution
85)

If the vectors \[a\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[-\mathbf{i}+5\mathbf{j}+a\mathbf{k}\] are perpendicular to each other, then \[a=\]          [MP PET 1996]

A)
            6

B)
            ? 6

C)
            5

D)
            ? 5
View Solution
86)

Which of the following is a true statement [Kurukshetra CEE 1996]

A)
            \[(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] is coplanar with c

B)
            \[(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] is perpendicular to a

C)
            \[(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] is perpendicular to b

D)
            \[(\mathbf{a}\times \mathbf{b})\times \mathbf{c}\] is perpendicular to c
View Solution
87)

If \[\mathbf{a}=\mathbf{i}-2\mathbf{j}\] and \[\mathbf{b}=2\mathbf{i}+\lambda \mathbf{j}\] are parallel, then \[\lambda \] is                                                 [RPET 1996]

A)
            4

B)
            2

C)
            ? 2

D)
            ? 4
View Solution
88)

If \[ai+6j-k\] and \[7i-3j+17k\] are perpendicular vectors, then the value of a is [Karnataka CET 2001]

A)
            5

B)
            ? 5

C)
            7

D)
            \[\frac{1}{7}\]
View Solution
89)

If \[4i+j-k\] and \[3i+mj+2k\] are at right angle, then \[m=\]                                     [Karnataka CET 2002]

A)
            ? 6

B)
            ? 8

C)
            ? 10

D)
            ? 12
View Solution
90)

If the vectors \[3i+\lambda \,j+k\] and \[2i-j+8k\] are perpendicular, then \[\lambda \] is [Kerala (Engg.) 2002]

A)
            ? 14

B)
            7

C)
            14

D)
            1/7
View Solution
91)

If a and b are two non-zero vectors, then the component of b along a is [MP PET 1991]

A)
            \[\frac{(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{a}}{\mathbf{b}\,.\,\mathbf{b}}\]

B)
            \[\frac{(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}}{\mathbf{a}\,.\,\mathbf{a}}\]

C)
            \[\frac{(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}}{\mathbf{a}\,.\,\mathbf{b}}\]

D)
            \[\frac{(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{a}}{\mathbf{a}\,.\,\mathbf{a}}\]
View Solution
92)

A vector of magnitude 14 lies in the xy-plane and makes an angle of \[{{60}^{o}}\] with x-axis. The components of the vector in the direction of x-axis and y-axis are

A)
            \[7,\,\,7\sqrt{3}\]

B)
            \[7\sqrt{3},\,\,7\]

C)
            \[14\sqrt{3},\,\,14/\sqrt{3}\]

D)
            \[14/\sqrt{3},\,\,14\sqrt{3}\]
View Solution
93)

If \[\mathbf{a}=4\mathbf{i}+6\mathbf{j}\] and \[\mathbf{b}=3\,\mathbf{j}+4\,\mathbf{k},\] then the component of a along b is [IIT Screening 1989; MNR 1983, 87; UPSEAT 2000]

A)
            \[\frac{18}{10\sqrt{3}}(3\mathbf{j}+4\mathbf{k})\]

B)
            \[\frac{18}{25}(3\mathbf{j}+4\mathbf{k})\]

C)
            \[\frac{18}{\sqrt{3}}(3\mathbf{j}+4\mathbf{k})\]

D)
            \[(3\mathbf{j}+4\mathbf{k})\]
View Solution
94)

Let \[\mathbf{b}=3\mathbf{j}+4\mathbf{k},\,\,\mathbf{a}=\mathbf{i}+\mathbf{j}\] and let \[{{\mathbf{b}}_{1}}\] and \[{{\mathbf{b}}_{2}}\] be component vectors of b parallel and perpendicular to a. If \[{{\mathbf{b}}_{1}}=\frac{3}{2}\mathbf{i}+\frac{3}{2}\mathbf{j}\], then \[{{\mathbf{b}}_{2}}=\] [MP PET 1989]

A)
            \[\frac{3}{2}\mathbf{i}+\frac{3}{2}\mathbf{j}+4\mathbf{k}\]

B)
            \[-\frac{3}{2}\mathbf{i}+\frac{3}{2}\mathbf{j}+4\mathbf{k}\]

C)
            \[-\frac{3}{2}\mathbf{i}+\frac{3}{2}\mathbf{j}\]

D)
            None of these
View Solution
95)

The component of \[\mathbf{i}+\mathbf{j}\] along \[\mathbf{j}+\mathbf{k}\] will be

A)
            \[\frac{\mathbf{i}+\mathbf{j}}{2}\]

B)
            \[\frac{\mathbf{j}+\mathbf{k}}{2}\]

C)
            \[\frac{\mathbf{k}+\mathbf{i}}{2}\]

D)
            None of these
View Solution
96)

The projection of vector \[2\mathbf{i}+3\mathbf{j}-2\mathbf{k}\] on the vector \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] will be      [RPET 1984, 90, 97, 99; Karnataka CET 2004]

A)
            \[\frac{1}{\sqrt{14}}\]

B)
            \[\frac{2}{\sqrt{14}}\]

C)
            \[\frac{3}{\sqrt{14}}\]

D)
            \[\sqrt{14}\]
View Solution
97)

If vector \[\mathbf{a}=2\mathbf{i}-3\mathbf{j}+6\mathbf{k}\] and vector \[\mathbf{b}=-2\mathbf{i}+2\mathbf{j}-\mathbf{k},\] then \[\frac{\text{Projection of vector }\mathbf{a}\text{ on vector }\mathbf{b}}{\text{Projection of vector }\mathbf{b}\text{ on vector }\mathbf{a}}=\]        [MP PET 1994, 99; Pb. CET 2000]

A)
            \[\frac{3}{7}\]

B)
            \[\frac{7}{3}\]

C)
            3

D)
            7
View Solution
98)

The projection of a along b is                          [RPET 1995]

A)
            \[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{a}|}\]

B)
            \[\frac{\mathbf{a}\,\times \,\mathbf{b}}{|\mathbf{a}|}\]

C)
            \[\frac{\mathbf{a}\,.\,\mathbf{b}}{|\mathbf{b}|}\]

D)
            \[\frac{\mathbf{a}\,\times \,\mathbf{b}}{|\mathbf{b}|}\]
View Solution
99)

If \[a=2i+j+2k\] and \[b=5i-3j+k,\] then the projection of b on a is [Karnataka CET 2002]

A)
            3

B)
            4

C)
            5

D)
            6
View Solution
100)

The projection of the vector \[i-2j+k\] on the vector \[4i-4j+7k\] is [RPET 1990; MNR 1980; MP PET 2002; UPSEAT 2002; Pb. CET 2004]

A)
            \[\frac{5\sqrt{6}}{10}\]

B)
            \[\frac{19}{9}\]

C)
            \[\frac{9}{19}\]

D)
            \[\frac{\sqrt{6}}{19}\]
View Solution

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JEE Main & Advanced Mathematics Vector Algebra Question Bank

done Scaler or Dot product of two vectors and its application Total Questions - 100

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Scaler or Dot product of two vectors and its application Total Questions - 100