Let O be the origin and O A → = 2 i ^ + 2 j ^ + k ^ , O B → = i ^ - 2 j ^ + 2 k ^ and O C → = 1 2 O B → - λ O A → for some λ > 0 . If O B → × O C → = 9 2 , then which of the following statements is (are) TRUE?

Area of the triangle A B C is 9 2

The acute angle between the diagonals of the parallelogram with adjacent sides O A → and O C → is π 3

Let a and b be positive real numbers. Suppose P Q → = a i ^ + b j ^ and P S → = a i ^ - b j ^ are adjacent sides of a parallelogram P Q R S . Let u → and v → be the projection vectors of w → = i ^ + j ^ along P Q → and P S → , respectively. If | u → | + v → = w → and if the area of the parallelogram P Q R S is 8 Sq. unit , then which of the following statements is/are TRUE?

The length of the diagonal P R → of the parallelogram P Q R S is 4 unit

w → is an angle bisector of the vectors P Q → and P S →

Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

If u ^ lies in the xy - plane then u 1 = u 2

There is exactly one choice for such v →

If u ^ lies in the xz - plane then 2 u 1 = u 3

Let O be the origin and O A → = 2 i ^ + 2 j ^ + k ^ , O B → = i ^ - 2 j ^ + 2 k ^ and O C → = 1 2 O B → - λ O A → for some λ > 0 . If O B → × O C → = 9 2 , then which of the following statements is (are) TRUE?

Area of the triangle A B C is 9 2

The acute angle between the diagonals of the parallelogram with adjacent sides O A → and O C → is π 3

Let a and b be positive real numbers. Suppose P Q → = a i ^ + b j ^ and P S → = a i ^ - b j ^ are adjacent sides of a parallelogram P Q R S . Let u → and v → be the projection vectors of w → = i ^ + j ^ along P Q → and P S → , respectively. If | u → | + v → = w → and if the area of the parallelogram P Q R S is 8 Sq. unit , then which of the following statements is/are TRUE?

The length of the diagonal P R → of the parallelogram P Q R S is 4 unit

w → is an angle bisector of the vectors P Q → and P S →

Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

If u ^ lies in the xy - plane then u 1 = u 2

There is exactly one choice for such v →

If u ^ lies in the xz - plane then 2 u 1 = u 3

HARD

JEE Advanced

IMPORTANT

Earn 100

Let $\vec{P R} = 3 \hat{i} + \hat{j} - 2 \hat{k} & \vec{S Q} = \hat{i} - 3 \hat{j} - 4 \hat{k}$ determine diagonals of a parallelogram $P Q R S$ and $\vec{P T} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ be another vector. Then the volume (in cubic unit) of the parallelepiped determined by the vectors $\vec{P T}, \vec{P Q} & \vec{P S}$ is

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Important Questions on Vector Algebra

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 2 - 2021>Q 1

Let

O

be the origin and

\vec{O A} = 2 \hat{i} + 2 \hat{j} + \hat{k}, \vec{O B} = \hat{i} - 2 \hat{j} + 2 \hat{k}

and

\vec{O C} = \frac{1}{2} (\vec{O B} - λ \vec{O A})

for some

λ > 0 .

|\vec{O B} \times \vec{O C}| = \frac{9}{2},

then which of the following statements is (are) TRUE?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 2 - 2020>Q 2

Let

a

and

b

be positive real numbers. Suppose

\vec{P Q} = a \hat{i} + b \hat{j}

and

\vec{P S} = a \hat{i} - b \hat{j}

are adjacent sides of a parallelogram

P Q R S .

Let

\vec{u}

and

\vec{v}

be the projection vectors of

\vec{w} = \hat{i} + \hat{j}

along

\vec{P Q}

and

\vec{P S}

, respectively. If

| \vec{u} | + |\vec{v}| = |\vec{w}|

and if the area of the parallelogram

P Q R S

8 Sq. unit,

then which of the following statements is/are TRUE?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 2 - 2019>Q 3

Let

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

and

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

be two vectors. Consider a vector

\vec{c} = α \vec{a} + β \vec{b}, α, β \in R .

If the projection of

\vec{c}

on the vector

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

3 \sqrt{2},

then the minimum value of

(\vec{c} - (\vec{a} \times \vec{b})) . \vec{c}

equals

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 2 - 2018>Q 4

Consider the cube in the first octant with sides

O P, O Q

and

O R

of length

1

, along the

x

-axis,

y

-axis and

z

-axis, respectively, where

O (0, 0, 0)

is the origin. Let

S (\frac{1}{2}, \frac{1}{2}, \frac{1}{2})

be the centre of the cube and

T

be the vertex of the cube opposite to the origin

O

such that

S

lies on the diagonal

O T

. If

\vec{p} = \vec{S P}, \vec{q} = \vec{S Q}, \vec{r} = \vec{S R}

and

\vec{t} = \vec{S T}

, then the value of

|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})|

is ______.

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 2 - 2016>Q 6

Let

\hat{u} = u_{1} \hat{i} + u_{2} \hat{j} + u_{3} \hat{k}

be a unit vector in

R^{3}

and

\hat{w} = \frac{1}{\sqrt{6}} (\hat{i} + \hat{j} + 2 \hat{k}) .

Given that there exists a vector

\vec{v}

R^{3}

such that

|\hat{u} \times \vec{v}| = 1

and

\hat{w} . (\hat{u} \times \vec{v}) = 1 .

Which of the following statement(s) is (are) correct ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2021>Q 10

Let

\vec{u}, \vec{v}

and

\vec{w}

be vectors in three-dimensional space, where

\vec{u}

and

\vec{v}

are unit vectors which are not perpendicular to each other and

\vec{u} \cdot \vec{w} = 1, \vec{v} \cdot \vec{w} = 1, \vec{w} \cdot \vec{w} = 4

.
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors

\vec{u}, \vec{v}

and

\vec{w}

, is

\sqrt{2}

, then the value of

| 3 \vec{u} + 5 \vec{v} |

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2020>Q 11

In a triangle $P Q R$ , let $\vec{a} = \vec{Q R}, \vec{b} = \vec{R P}$ and $\vec{c} = \vec{P Q}$ .

If $|\vec{a}| = 3, |\vec{b}| = 4$ and $\frac{\vec{a} . (\vec{c} - \vec{b})}{\vec{c} . (\vec{a} - \vec{b})} = \frac{|\vec{a}|}{|\vec{a}| + |\vec{b}|}$ , then the value of ${|\vec{a} \times \vec{b}|}^{2}$ is _______

MEDIUM

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 18 - Vector Algebra>JEE Advanced Paper 1 - 2018>Q 12

Let

\vec{a}

and

\vec{b}

be two unit vectors such that

\vec{a} . \vec{b} = 0 .

For some

x, y \in R,

let

\vec{c} = x \vec{a} + y \vec{b} + (\vec{a} \times \vec{b}) .

|\vec{c}| = 2

and the vector

\vec{c}

is inclined at the same angle

α

to both

\vec{a}

and

\vec{b},

then the value of

8 \cos^{2} α

Let PR→=3i⁡^+j⁡^-2k^&SQ→=i⁡^-3j⁡^-4k^ determine diagonals of a parallelogram PQRS and PT→=i⁡^+2j⁡^+3k^ be another vector. Then the volume (in cubic unit) of the parallelepiped determined by the vectors PT→, PQ→ & PS→ is

Important Questions on Vector Algebra

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