The unit vector perpendicular to ${\bf{\hat i}} - {\bf{\hat j}}$ and coplanar with ${\bf{\hat i}} + 2{\bf{\hat j}}{\rm{\;and\;}}2{\bf{\hat i}} + 3{\bf{\hat j}}$ is

$\frac{1}{{\sqrt 2 }}\left( {{\bf{\hat i}} + {\bf{\hat j}}} \right)$

$\frac{{2{\bf{\hat i}} - 5{\bf{\hat j}}}}{{\sqrt {2a} }}$

$2{\bf{\hat i}} + 5{\bf{\hat j}}$

${\bf{\hat i}} + {\bf{\hat j}}$

A unit vector in the plane of $\widehat {\bf{i}} + 2\widehat {\bf{j}} + {\bf{\hat k}}$ and $\widehat {\bf{i}} + \widehat {\bf{j}} + 2{\bf{\hat k}}$ and perpendicular to 2$\widehat {\bf{i}} + \widehat {\bf{j}} + {\bf{\hat k}}$ is

$\frac{{\widehat {\bf{j}} - {\bf{\hat k}}}}{{\sqrt 2 }}$

$\widehat {\bf{j}} - {\bf{\hat k}}$

$\frac{{\widehat {\bf{i}} + \widehat {\bf{j}}}}{{\sqrt 2 }}$

$\frac{{\widehat {\bf{j}} + {\bf{\hat k}}}}{{\sqrt 2 }}$

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JEE • MATHEMATICS • DOT PRODUCT OF VECTORS

Dot product of vectors practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 61-90 of 296 questions

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61 Easy

61 of 296

Analyzing +4 | -1

If the angle between the vectors

\mathbf{a}=\hat{\mathbf{i}}+(\cos x) \hat{\mathbf{j}}+\hat{\mathbf{k}}

and

\mathbf{b}=\left(\sin ^2 x-\sin x\right) \hat{\mathbf{i}}-(\cos x) \hat{\mathbf{j}}+(3-4 \sin x) \hat{\mathbf{k}}

is obtuse and

x \in\left(0, \frac{\pi}{2}\right)

, then the exhaustive set of values of '

x

' is equal to

View answer and explanation

Correct option: B

x \in\left(\frac{\pi}{6}, \frac{\pi}{2}\right)

Option B (

x \in (\frac{\pi}{6}, \frac{\pi}{2})

) is correct because the angle between the vectors is obtuse, implying their dot product is negative, which leads to the solution

x \in (\frac{\pi}{6}, \frac{\pi}{2})

62 Easy

62 of 296

Application +4 | -1

The angle between

{a}+2 {b}

and

{a}-3 {b}

|{a}|

=1,|{b}|=2

and angle between

{a}

and

{b}

60^{\circ}

View answer and explanation

Correct option: B

\cos ^{-1} \frac{-24}{\sqrt{21} \sqrt{31}}

Option B (

\cos ^{-1} \frac{-24}{\sqrt{21} \sqrt{31}}

) is correct because the angle between the vectors

{a}+2 {b}

and

{a}-3 {b}

is calculated using the dot product formula and the given magnitudes and angle between {a} and {b}.

63 Easy

63 of 296

Application +4 | -1

The summation of two unit vectors is a third unit vector, then the modulus of the difference of the unit vectors is

View answer and explanation

Correct option: A

\sqrt 3

Option A (

\sqrt 3

) is correct because the magnitude of the difference of two unit vectors, given that their sum is also a unit vector, can be derived using vector algebra.

64 Easy

64 of 296

Analyzing +4 | -1

\mathbf{\vec a} + \mathbf{\vec b} + \mathbf{\vec c} = \vec 0

and

\left| {\mathbf{\vec a}} \right| = 3,\left| {\mathbf{\vec b}} \right| = 5,\mathbf{\vec a} + \left| {\mathbf{\vec c}} \right| = 7,

then angle between

\mathbf{\vec a}\mathrm{\;and\;}\mathbf{\vec b}

View answer and explanation

Correct option: B

\frac{\pi }{3}

Option B (

\frac{\pi }{3}

) is correct because it is the angle between the vectors that satisfies the given vector equation and magnitudes.

65 Easy

65 of 296

Understanding +4 | -1

If a is any vector in space, then

View answer and explanation

Correct option: A

\mathbf{a}=(\mathbf{a} \cdot \mathbf{i}) \mathbf{i}+(\mathbf{a} . \mathbf{j}) \mathbf{j}+(\mathbf{a} \cdot \mathbf{k}) \mathbf{k}

Option A (

\mathbf{a}=(\mathbf{a} \cdot \mathbf{i}) \mathbf{i}+(\mathbf{a} . \mathbf{j}) \mathbf{j}+(\mathbf{a} \cdot \mathbf{k}) \mathbf{k}

) is correct because it represents the decomposition of vector

\mathbf{a}

into its components along the coordinate axes.

66 Medium

66 of 296

Application +4 | -1

Constant forces

{{\mathbf{\vec P}}_1} = \widehat \mathbf{i} - \widehat \mathbf{j} + \mathbf{\hat k}\;

{{\mathbf{\vec P}}_2} = - \widehat \mathbf{i} + 2\widehat \mathbf{j} - \mathbf{\hat k}

and

{{\mathbf{\vec P}}_3} = \widehat \mathbf{j} - \mathbf{\hat k}

act on a particle at point

A

. The work done when the particle is displaced from the point

A

B

where

{{\mathbf{\vec A}}_\;} = 4\widehat \mathbf{i} - 3\widehat \mathbf{j} - 2\mathbf{\hat k}

and

{{\mathbf{\vec B}}_\;} = 6\widehat \mathbf{i} + \widehat \mathbf{j} - 3\mathbf{\hat k}

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the work done calculated using the dot product of the total force and displacement is 8, which is not among the given options.

67 Easy

67 of 296

Application +4 | -1

The distance of the point

B

with position vector

{i}+2 {j}+3{k}

from the line passing through the point

A

with position vector

4 {i}+2 {j}+2 {k}

and parallel to the vector

2 {i}+3 {j}+6 {k}

View answer and explanation

Correct option: A

\sqrt{10}

Option A (

\sqrt{10}

) is correct because the distance is found by recognizing that AB is perpendicular to the line, making the distance simply |AB|.

68 Easy

68 of 296

Application +4 | -1

The unit vector perpendicular to

\mathbf{\hat i} - \mathbf{\hat j}

and coplanar with

\mathbf{\hat i} + 2\mathbf{\hat j}\mathrm{\;and\;}2\mathbf{\hat i} + 3\mathbf{\hat j}

View answer and explanation

Correct option: C

\frac{1}{{\sqrt 2 }}\left( {\mathbf{\hat i} + \mathbf{\hat j}} \right)

Option C (

\frac{1}{\sqrt 2 }(\mathbf{\hat i} + \mathbf{\hat j})

) is correct because it is a unit vector perpendicular to

\mathbf{\hat i} - \mathbf{\hat j}

and coplanar with the other given vectors.

69 Easy

69 of 296

Application +4 | -1

Let

\vec b = 4\hat i + 3\hat j

be two vectors perpendicular to each other in the

xy

-plane. Then, a vector in the same plane having projections 1 and 2 along

{\vec b}

and

{\vec c}

, respectively, is

View answer and explanation

Correct option: B

2\hat i - \hat j

Option B (

2\hat i - \hat j

) is correct because it satisfies the conditions of being in the xy-plane and having projections 1 and 2 along the given vectors.

70 Medium

70 of 296

Analyzing +4 | -1

Let

\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}-\hat{j}+2 \hat{k}

and

\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}

be three vectors if

\vec{r}

is a vector such that,

\vec{r} \times \vec{b}=\vec{c} \times \vec{b}

and

\vec{r} \cdot \vec{a}=0

, then

25|\vec{r}|^{2}

is equal to

View answer and explanation

Correct option: D

339

Option D (339) is correct because by solving the vector equation and using the dot product condition, we find that

25|\vec{r}|^{2} = 339

71 Easy

71 of 296

Application +4 | -1

The distance between the line

\mathbf{r}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}})

and the plane

\mathbf{r} \cdot(\hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}})=5

View answer and explanation

Correct option: C

\frac{10}{3 \sqrt{3}}

Option C (

\frac{10}{3 \sqrt{3}}

) is correct because the line is parallel to the plane, and the distance is calculated using the formula for the distance between a point on the line and the plane.

72 Medium

72 of 296

Analyzing +4 | -1

\vec{a}=\hat{i}+2 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=7 \hat{i}-3 \hat{j}+4 \hat{k}

\vec{r} \times \vec{b}+\vec{b} \times \vec{c}=\overrightarrow{0}

and

\vec{r} \cdot \vec{a}=0

. Then

\vec{r} \cdot \vec{c}

is equal to

View answer and explanation

Correct option: C

Option C (34) is correct because it is the value obtained by correctly solving for vector r using the given conditions and then calculating the dot product.

73 Easy

73 of 296

Application +4 | -1

A particle acted on by constant forces

4 \hat{i}+\hat{j}-3 \hat{k}

and

3 \hat{i}+\hat{j}-\hat{k}

is displaced from the point

\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}

to the point

5 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\hat{\mathrm{k}}

. The total work done by the forces is

View answer and explanation

Correct option: D

40 units.

Option D (40 units) is correct because the work done is the dot product of the total force and the displacement vector.

74 Easy

74 of 296

Application +4 | -1

Let

{a}={i}+{j}+{k}, {b}={i}-{j}+{k}

and

{c}={i}-{j}-{k}

be three vectors. A vector

{v}

in the plane of

{a}

and

{b}

, whose projection on

{c}

1 / \sqrt{3}

, is given by

View answer and explanation

Correct option: C

3 {i}-{j}+{k}

Option C (3i - j + k) is correct because it lies in the plane of a and b, and its projection on c is 1/

\sqrt{3}

75 Easy

75 of 296

Analyzing +4 | -1

Let a, b,c, be vectors of length

3,4,5

respectively and

\mathbf{a}

be perpendicular to

(\mathbf{b}+\mathbf{c}), \mathbf{b}

(\mathbf{c}+\mathbf{a})

and

\mathbf{c}

(\mathbf{a}+\mathbf{b})

, then the value of

(\mathbf{a}+\mathbf{b}+\mathbf{c})

View answer and explanation

Correct option: D

5 \sqrt{2}

The correct answer is D (

5 \sqrt{2}

) because the magnitude of the sum of the vectors is calculated using the given perpendicularity conditions and vector lengths.

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76 Easy

76 of 296

Application +4 | -1

Let

\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-\hat{k}

and

\vec{c}=\hat{i}+\hat{j}-2 \hat{k}

be three vectors. A vector of the type

\vec{b}+\lambda \vec{c}

for some scalar

\lambda

, whose projection on

\vec{a}

is of magnitude

\sqrt{\frac{2}{3}}

is :

View answer and explanation

Correct option: B

2 \hat{i}+3 \hat{j}-3 \hat{k}

Option B (

2 \hat{i}+3 \hat{j}-3 \hat{k}

) is correct because by finding the vector

\vec{d} = \vec{b} + \lambda \vec{c}

and then using the projection formula, we can solve for

\lambda

and find the vector.

77 Easy

77 of 296

Application +4 | -1

\theta

be the angle between the vectors

a=2 i+2 j-k

and

\mathbf{b}=6 \mathbf{i}-3 \mathbf{j}+2 \mathbf{k}

, then

View answer and explanation

Correct option: A

\cos \theta=\frac{4}{21}

Option A (

\cos \theta=\frac{4}{21}

) is correct because it is the result of applying the dot product formula to find the cosine of the angle between the two vectors.

78 Easy

78 of 296

Analyzing +4 | -1

A unit vector in the plane of

\widehat \mathbf{i} + 2\widehat \mathbf{j} + \mathbf{\hat k}

and

\widehat \mathbf{i} + \widehat \mathbf{j} + 2\mathbf{\hat k}

and perpendicular to 2

\widehat \mathbf{i} + \widehat \mathbf{j} + \mathbf{\hat k}

View answer and explanation

Correct option: D

\frac{{\widehat \mathbf{j} - \mathbf{\hat k}}}{{\sqrt 2 }}

Option D (

\frac{{\widehat \mathbf{j} - \mathbf{\hat k}}}{{\sqrt 2 }}

) is correct because it represents a unit vector in the plane of the given vectors and is perpendicular to the specified vector.

79 Medium

79 of 296

Analyzing +4 | -1

The value of '

a

' for which the straight line

4 x-3 y+4 z-2=0=3 x-2 y+z-5

is parallel to the plane

2 x-y+a z-7=0

, is:

View answer and explanation

Correct option: B

-2

Option B (-2) is correct because the dot product of the parallel vector of the line and the normal vector of the plane must be zero for the line to be parallel to the plane.

80 Easy

80 of 296

Application +4 | -1

\hat{x}, \hat{y}

and

\hat{z}

are three unit vectors in three-dimensional space, then the minimum value of

|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2

View answer and explanation

Correct option: B

Option B (3) is correct because the minimum value is found by using the inequality

(\hat{x}+\hat{y}+\hat{z})^2 \geq 0

and manipulating the given expression.

81 Easy

81 of 296

Application +4 | -1

\left| {\;\mathbf{\vec a} + \mathbf{\vec b}\;\left| = \right|\mathbf{\vec a} - \mathbf{\vec b}\mathrm{\;}} \right|

, then angle between

\mathbf{\vec a}\mathrm{\;and\;}\mathbf{\vec b}\mathrm{\;}

\left( {\;\mathbf{\vec a}\; \ne \vec 0,\mathbf{\vec b} \ne \vec 0} \right)

View answer and explanation

Correct option: D

\frac{\pi }{2}

Option D (

\frac{{\pi }}{{2}}

) is correct because if the magnitudes of the sum and difference of two vectors are equal, the angle between them is 90 degrees.

82 Easy

82 of 296

Application +4 | -1

Let

\overrightarrow{\mathrm{a}}

and

\overrightarrow{\mathrm{b}}

be two unit vectors such that

|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|=\sqrt{3}

.
If

\vec{c}=\vec{a}+2 \vec{b}+3(\vec{a} \times \vec{b})

, then

2|\vec{c}|

is equal to :

View answer and explanation

Correct option: A

\sqrt{55}

Option D (

12 \sqrt{2}

) is correct because it is the value of the dot product

\overrightarrow{\mathrm{c}} \cdot(-7 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})

calculated using the given conditions.

83 Easy

83 of 296

Application +4 | -1

Let

\mathbf{\vec a}\mathrm{\;}\,\,\mathrm{and}\,\mathrm{\;}\mathbf{\vec b}

be unit vectors inclined at an angle

2\alpha \,\,\left( {0 \le \alpha \le \pi } \right)

each other, then

\left| {\;\,\mathbf{\vec a} + \mathbf{\vec b}\,} \right| < 1

, if

View answer and explanation

Correct option: D

\frac{\pi }{3} < \alpha < \frac{{2\,\pi }}{3}

Option D (

\frac{\pi}{3} < \alpha < \frac{{2\pi}}{3}

) is correct because it represents the range of

\alpha

for which the magnitude of the sum of the unit vectors is less than 1.

84 Medium

84 of 296

Application +4 | -1

The unit vector which is orthogonal to the vector

3\widehat i + 2\widehat j + 6\widehat k

and is coplanar with the vectors

2\widehat i + \widehat j + \widehat k

and

\widehat i - \widehat j + \widehat k

View answer and explanation

Correct option: C

\frac{{3\widehat j - \widehat k}}{{\sqrt {10} }}

Option C (

\frac{{3\widehat j - \widehat k}}{{\sqrt {10} }}

) is correct because it represents a unit vector orthogonal to the given vector and coplanar with the other two vectors, derived using vector triple product.

85 Easy

85 of 296

Understanding +4 | -1

The vector equation of the sphere whose centre is the point (1,0,1)and radius is 4, is

View answer and explanation

Correct option: A

\left| {\mathbf{\vec r} - \left( {\widehat \mathbf{i} + \mathbf{\hat k}} \right)} \right| = 4

Option A (

\left| {\mathbf{\vec r} - \left( {\widehat {\mathbf{i}} + \mathbf{\hat k}} \right)} \right| = 4

) is correct because the vector equation of a sphere with center

\vec a

and radius R is given by

\left| {\mathbf{\vec r} - \mathbf{\vec a}} \right| = R

86 Easy

86 of 296

Application +4 | -1

If a has magnitude 5 and points North-East and vector

\mathbf{b}

has magnitude 5 and points North-West, then

|\mathbf{a}-\mathbf{b}|

is equal to

View answer and explanation

Correct option: D

5 \sqrt{2}

Option D (

5 \sqrt{2}

) is correct because it is the magnitude of the difference between the two vectors, calculated using the Pythagorean theorem since the vectors are perpendicular.

87 Easy

87 of 296

Application +4 | -1

If the vectors

a \hat{i}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}

and

-\hat{\mathrm{i}}+5 \hat{\mathrm{j}}+\mathrm{a} \hat{\mathrm{k}}

are perpendicular to each other then

a

is equal to:

View answer and explanation

Correct option: C

-5

Option C (-5) is correct because the dot product of two perpendicular vectors is zero, leading to the equation -a + 10 + 3a = 0, which simplifies to a = -5.

88 Easy

88 of 296

Application +4 | -1

\vec a + \vec b + \vec c = 0,\;\left| {\vec a} \right| = 3,\;\left| {\vec b} \right| = 5,\;\left| {\vec c} \right| = 7

, then the angle between

{\vec a}

and

{\vec b}

View answer and explanation

Correct option: A

\pi /3

Option A (

\pi /3

) is correct because the angle between the vectors is found using the cosine rule derived from the given vector equation.

89 Easy

89 of 296

Analyzing +4 | -1

\mathbf{a}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}

and

\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}

are given vector.

A

vector

\mathrm{w}

which is perpendicular to

Z

-axis satisfying

\mathbf{c} \cdot \mathbf{a}=9

and

\mathbf{c} \cdot \mathbf{b}=-4

. If inclination of

\mathbf{c}

with

X

-axis and

Y

-axis is

\alpha

and

\beta

respectively, then which of the following is not true?

View answer and explanation

Correct option: C

\alpha>\frac{\pi}{2}

The correct answer is C (

\alpha>\frac{\pi}{2}

) because based on the calculated vector c, the angle alpha is less than

\frac{\pi}{2}

90 Easy

90 of 296

Application +4 | -1

If the constant force

2\widehat \mathbf{i} - 5\widehat \mathbf{j} + 6\mathbf{\hat k}

and

- \widehat \mathbf{i} + 2\widehat \mathbf{j} - \mathbf{\hat k}

act on a particle due to which it is displaced from a point

A\left( {4, - 3, - 2} \right)

to a point

B\left( {6,1, - 3} \right)

then the work done by the force is

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the work done is calculated as -15 units, which is not among the given options.

Revision notes and key takeaways

Review DOT PRODUCT OF VECTORS in your textbook or class notes, then use this set to check your understanding.

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