JEE Main > Vectors

Explore popular questions from Vectors for JEE Main. This collection covers Vectors previous year JEE Main questions hand picked by popular teachers.


Q 1.    

Correct4

Incorrect-1

If {tex} | \vec { a } | = 4 , | \vec { b } | = 2 {/tex} and the angle between {tex} \vec { a } {/tex} and {tex} \vec { b } {/tex} is {tex} \frac { \pi } { 6 } {/tex} then {tex} ( \vec { a } \times \vec { b } ) ^ { 2 } {/tex} is equal to

A

48

16

C

{tex} \vec { a } {/tex}

D

none of these

Explanation


Q 2.    

Correct4

Incorrect-1

If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are vectors such that {tex} [ \vec { a } \quad \vec { b } \quad \vec { c } ] = 4 {/tex}
then {tex} \left[ \begin{array} { l l l } { \vec { a } \times \vec { b } } & { \vec { b } \times \vec { c } } & { \vec { c } \times \vec { a } } \end{array} \right] = {/tex}

16

B

64

C

4

D

8

Explanation


Q 3.    

Correct4

Incorrect-1

If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are vectors show that {tex} \vec { a } + \vec { b } + \vec { c } = 0 {/tex} and {tex} | \vec { a } | = 7 , | \vec { b } | = 5 , | \vec { c } | = 3 {/tex} then angle between vector {tex} \vec { b } {/tex} and {tex} \vec { c } {/tex} is

{tex} 60 ^ { \circ } {/tex}

B

{tex} 30 ^ { \circ } {/tex}

C

{tex} 45 ^ { \circ } {/tex}

D

{tex} 90 ^ { \circ } {/tex}

Explanation


Q 4.    

Correct4

Incorrect-1

If {tex} | a | = 5 , | b | = 4 , | c | = 3 {/tex} thus what will be the value {tex} \mathrm { of } \quad | a \cdot b + b \cdot c + c \cdot a | , {/tex} given that {tex} \vec { a } + \vec { b } + \vec { c } = 0 {/tex}

25

B

50

C

- 25

D

- 50

Explanation



Q 5.    

Correct4

Incorrect-1

{tex} 3 \lambda \vec { c } + 2 \mu ( \vec { a } \times \vec { b } ) = 0 {/tex} then

A

{tex} 3 \lambda + 2 \mu = 0 {/tex}

{tex} 3 \lambda = 2 \mu {/tex}

C

{tex} \lambda = \mu {/tex}

D

{tex} \lambda + \mu = 0 {/tex}

Explanation


Q 6.    

Correct4

Incorrect-1

{tex} \vec { a } = 3 \hat { i } - 5 \hat { j } {/tex} and {tex} \vec { b } = 6 \hat { i } + 3 \hat { j } {/tex} are two vectors and {tex} \vec { c } {/tex} is a vector such that {tex} \vec { c } = \vec { a } \times \vec { b } {/tex} then {tex} | \vec { a } |: | \vec { b } |: | \vec { c } | = {/tex}

A

{tex} \sqrt { 34 }: \sqrt { 45 }: \sqrt { 39 } {/tex}

{tex} \sqrt { 34 }: \sqrt { 45 }: 39 {/tex}

C

{tex}34:39:45{/tex}

D

{tex}39:35:34{/tex}

Explanation



Q 7.    

Correct4

Incorrect-1

If {tex} \vec { a } \times \vec { b } = \vec { b } \times \vec { c } = \vec { c } \times \vec { a } {/tex} then {tex} \vec { a } + \vec { b } + \vec { c } = {/tex}

A

abc

B

-1

0

D

2

Explanation


Q 8.    

Correct4

Incorrect-1

Consider {tex} A , B , C {/tex} and {tex} D {/tex} with position vectors {tex} 7 \hat { i } - 4 \hat { j } + 7 \hat { k } , \quad \hat { i } - 6 \hat { j } + 10 \hat { k } , \quad - \hat { i } - 3 \hat { j } + 4 \hat { k } \quad {/tex} and {tex} 5 \hat { i } - \hat { j } + 5 \hat { k } {/tex} respectively. Then {tex} A B C D {/tex} is a

A

rhombus

B

rectangle

C

parallelogram but not a rhombus

None of these

Explanation


Q 9.    

Correct4

Incorrect-1

If {tex} \vec { u } , \vec { v } {/tex} and {tex} \vec { w } {/tex} are three non-coplanar vectors, then {tex} ( \vec { u } + \vec { v } - \vec { w } ) \cdot ( \vec { u } - \vec { v } ) \times ( \vec { v } - \vec { w } ) {/tex} equals

{tex} \vec { u } \cdot (\vec { v } \times \vec { w } ){/tex}

B

{tex} \vec { u } \cdot \vec { w } \times \vec { v } {/tex}

C

{tex} 3 u \cdot \vec { u } \times \vec { w } {/tex}

D

0

Explanation


Q 10.    

Correct4

Incorrect-1

Let {tex} \vec { u } = \hat { i } + \hat { j } , \vec { v } = \hat { i } - \hat { j } {/tex} and {tex} \vec { w } = \hat { i } + 2 \hat { j } + 3 \hat { k } . {/tex} If {tex} \hat { n } {/tex} is a unit vector such that {tex} \vec { u } \cdot \hat { n } = 0 {/tex} and {tex} \vec { v } \cdot \hat { n } = 0 {/tex}, then {tex} | \vec { w } \cdot \hat { n } | {/tex} is equal to

A

1

B

2

3

D

0

Explanation

Q 11.    

Correct4

Incorrect-1

The vectors {tex} \overrightarrow { A B } = 3 \hat { i } + 4 \hat { k } {/tex} and {tex} \overrightarrow { A C } = 5 \hat { i } - 2 \hat { j } + 4 \hat { k } {/tex} are the sides of a triangle {tex} A B C {/tex}. The length of the median through {tex} A {/tex} is

A

{tex} \sqrt { 72 } {/tex}

{tex} \sqrt { 33 } {/tex}

C

{tex} \sqrt { 288 } {/tex}

D

{tex} \sqrt { 18 } {/tex}

Explanation


Q 12.    

Correct4

Incorrect-1

{tex} \vec { a } , \vec { b } , \vec { c } {/tex} are three vectors, such that {tex} \vec { a } + \vec { b } + \vec { c } = 0 , {/tex} {tex} | \vec { b } | = 2 , | \vec { c } | = 3 , \quad {/tex} then {tex} \quad \vec { a } \cdot \vec { b } + \vec { b } \cdot \vec { c } + \vec { c } \cdot \vec { a } \quad {/tex} is equal to

- 7

B

7

C

1

D

0

Explanation

Q 13.    

Correct4

Incorrect-1

Let {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} be three non-zero vectors such that no two of these are collinear. If the vector {tex} \vec { a } + 2 \vec { b } {/tex} is collinear with {tex} \vec { c } {/tex} and {tex} \vec { b } + 3 \vec { c } {/tex} is collinear with {tex} \vec { a } {/tex} ( {tex}\lambda {/tex} being some non-zero scalar) then {tex} \vec { a } + 2 \vec { b } + 6 \vec { c } {/tex} equals

A

{tex} \lambda \vec { c } {/tex}

B

{tex} \lambda \vec { b } {/tex}

C

{tex} \lambda \vec { a } {/tex}

0

Explanation


Q 14.    

Correct4

Incorrect-1

A particle is acted upon by constant forces {tex} 4 \hat { i } + \hat { j } - 3 \hat { k } {/tex} and {tex} 3 \hat { i } + \hat { j } - \hat { k } {/tex} which displace it from a point {tex} \hat { i } + 2 \hat { j } + 3 \hat { k } {/tex} to the point {tex} 5 \hat { i } + 4 \hat { j } + \hat { k } {/tex}. The work done in standard units by the forces is given by

A

25

B

30

40

D

15

Explanation


Q 15.    

Correct4

Incorrect-1

If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are non-coplanar vectors and {tex} \lambda {/tex} is a real number, then the vectors {tex} \vec { a } + 2 \vec { b } + 3 \vec { c } , \lambda \vec { b } + 4 \vec { c } {/tex} and {tex} ( 2 \lambda - 1 ) \vec { c } {/tex} are non-coplanar for

all except two values of {tex} \lambda {/tex}

B

all except one value of {tex} \lambda {/tex}

C

all values of {tex} \lambda {/tex}

D

no value of {tex} \lambda {/tex}

Explanation


Q 16.    

Correct4

Incorrect-1

Let {tex} \vec { u } , \vec { v } , \vec { w } {/tex} be such that {tex} | \vec { u } | = 1 , | \vec { v } | = 2 {/tex}, {tex} | \vec { w } | = 3 . {/tex} If the projection {tex} \vec { v } {/tex} along {tex} \vec { u } {/tex} is equal to that of {tex} \vec { w } {/tex} along {tex} \vec { u } {/tex} and {tex} \vec { v } , \vec { w } {/tex} are perpendicular to each other then {tex} | \vec { u } - \vec { v } + \vec { w } | {/tex} equals

{tex} \sqrt { 14 } {/tex}

B

{tex} \sqrt { 7 } {/tex}

C

{tex}2{/tex}

D

{tex}14{/tex}

Explanation


Q 17.    

Correct4

Incorrect-1

Let {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} be non-zero vectors such that {tex} ( \vec { a } \times \vec { b } ) \times \vec { c } = \frac { 1 } { 3 } | \vec { b } | | \vec { c } | \vec { a }{/tex} . If {tex} \theta {/tex} is the acute angle between the vectors {tex} b {/tex} and {tex} \vec { c } , {/tex} then {tex} \sin \theta {/tex} equals

A

{tex} \frac { 2 } { 3 } {/tex}

B

{tex} \frac { \sqrt { 2 } } { 3 } {/tex}

C

{tex} \frac { 1 } { 3 } {/tex}

{tex} \frac { 2 \sqrt { 2 } } { 3 } {/tex}

Explanation


Q 18.    

Correct4

Incorrect-1

If {tex} C {/tex} is the mid point of {tex} A B {/tex} and {tex} P {/tex} is any point outside {tex} A B , {/tex} then

A

{tex} \overrightarrow { P A } + \overrightarrow { P B } + \overrightarrow { P C } = 0 {/tex}

B

{tex} \overrightarrow { P A } + \overrightarrow { P B } + 2 \overrightarrow { P C } = \overrightarrow { 0 } {/tex}

C

{tex} \overrightarrow { P A } + \overrightarrow { P B } = \overrightarrow { P C } {/tex}

{tex} \overrightarrow { P A } + \overrightarrow { P B } = 2 \overrightarrow { P C } {/tex}

Explanation



Q 19.    

Correct4

Incorrect-1

For any vector {tex} \vec { a } , {/tex} the value of {tex} ( \vec { a } \times \hat { i } ) ^ { 2 } + ( \vec { a } \times \hat { j } ) ^ { 2 } + ( \vec { a } \times \hat { k } ) ^ { 2 } {/tex} is equal to

A

{tex} \vec { a } ^ { 2 } {/tex}

B

{tex} 3 \vec { a } ^ { 2 } {/tex}

C

{tex} 4 \vec { a } ^ { 2 } {/tex}

{tex} 2 \vec { a } ^ { 2 } {/tex}

Explanation


Q 20.    

Correct4

Incorrect-1

Let {tex} \vec { a } = \hat { i } - \hat { k } , \vec { b } = x \hat { i } + \hat { j } + ( 1 - x ) \hat { k } {/tex} and {tex} \vec { c } = y \hat { i } + x \hat { j } + ( 1 + x - y ) \hat { k } {/tex}.
Then {tex} [ \vec { a } , \vec { b } , \vec { c } ] {/tex} depends on

A

only {tex} x {/tex}

B

only {tex} y {/tex}

neither {tex} x {/tex} nor {tex} y {/tex}

D

both {tex} x {/tex} and {tex} y {/tex}

Explanation



Q 21.    

Correct4

Incorrect-1

Let {tex} a , b {/tex} and {tex} c {/tex} be distinct non-negative numbers. If the vectors {tex} a \hat { i } + a \hat { j } + c \hat { k } , \hat { i } + \hat { k } {/tex} and {tex} c \hat { i } + \hat { c } \hat { j } + b \hat { k } {/tex} lie in a plane, then {tex} c {/tex} is

A

the arithmetic mean of {tex} a {/tex} and {tex} b {/tex}

the geometric mean of {tex} a {/tex} and {tex} b {/tex}

C

the harmonic mean of {tex} a {/tex} and {tex} b {/tex}

D

equal to zero.

Explanation


Q 22.    

Correct4

Incorrect-1

If {tex} \vec { a } , \vec { b } , \vec { c } {/tex} are non-coplanar vector and {tex} \lambda {/tex} is a real number then {tex} \left[ \lambda ( \vec { a } + \vec { b } ) \lambda ^ { 2 } \vec { b } \lambda \vec { c } \right] = \left[ \begin{array} { c c c } { \vec { a } } & { \vec { b } + \vec { c } } & { \vec { b } } \end{array} \right] {/tex} for

no value of {tex} \lambda {/tex}

B

exactly one value of {tex} \lambda {/tex}

C

exactly two values of {tex} \lambda {/tex}

D

exactly three values of {tex} \lambda {/tex}

Explanation


Q 23.    

Correct4

Incorrect-1

If {tex} ( \vec { a } \times \vec { b } ) \times \vec { c } = \vec { a } \times ( \vec { b } \times \vec { c } ) , {/tex} where {tex} \vec { a } , \vec { b } {/tex} and {tex} \vec { c } {/tex} are any three vectors such that {tex} \vec { a } \cdot \vec { b } \neq 0 , \vec { b } \cdot \vec { c } \neq 0 , {/tex} then {tex} \vec { a } {/tex} and {tex} \vec { c } {/tex} are

A

inclined at an angle of {tex} \pi / 3 {/tex} between them

B

inclined at an angle of {tex} \pi / 6 {/tex} between them

C

perpendicular

parallel

Explanation


Q 24.    

Correct4

Incorrect-1

The values of a, for which the points A, B, C with position vectors {tex} 2\vec{i} - \vec{j} + \vec{k}, \vec{i} - 3\vec{j}-5\vec{k} {/tex} and {tex} a \vec{i} - 3 \vec{j} + \vec{k} {/tex} respectively are the vertices of a right­-angled triangle at {tex}c{/tex} are

2 and 1

B

- 2 and - 1

C

- 2 and 1

D

2 and - 1

Explanation


Q 25.    

Correct4

Incorrect-1

If {tex} \hat { u } {/tex} and {tex} \hat { v } {/tex} are unit vectors and {tex} \theta {/tex} is the acute angle between them, then {tex} 2 \hat { u } \times 3 \hat { v } {/tex} is a unit vector for

A

no value of {tex} \theta {/tex}

exactly one value of {tex} \theta {/tex}

C

exactly two values of {tex} \theta {/tex}

D

more than two values of {tex} \theta {/tex}

Explanation