Let a → = 5 i ^ - j ^ - 3 k ^ and b → = i ^ + 3 j ^ + 5 k ^ be two vectors. Then which one of the following statements is TRUE?

Projection of a → on b → is - 13 35 and the direction of the projection vector is opposite to the direction of b →

Projection of a → on b → is - 17 35 and the direction of the projection vector is opposite to the direction of b →

Projection of a → on b → is 17 35 and the direction of the projection vector is opposite to the direction of b →

Projection of a → on b → is 13 35 and the direction of the projection vector is opposite to the direction of a →

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IMPORTANT

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Let $\vec{a} = 2 \hat{i} - \hat{j} + 5 \hat{k}$ and $\vec{b} = α \hat{i} + β \hat{j} + 2 \hat{k}$ . If $((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k} = \frac{23}{2}$ , then $|\vec{b} \times 2 \hat{j}|$ is equal to

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Important Questions on Vector Algebra

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 27 July 2022 Shift 2>Q 110

Let

\vec{a}, \vec{b}, \vec{c}

be three non-coplanar vectors such that

\vec{a} \times \vec{b} = 4 \vec{c}, \vec{b} \times \vec{c} = 9 \vec{a}

and

\vec{c} \times \vec{a} = α \vec{b}, α > 0

|\vec{a}| + |\vec{b}| + |\vec{c}| = 36

, then

α

is equal to _______.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 July 2022 Shift 1>Q 111

Let the vectors

\vec{a} = (1 + t) \hat{i} + (1 - t) \hat{j} + \hat{k}

\vec{b} = (1 - t) \hat{i} + (1 + t) \hat{j} + 2 \hat{k}

and

\vec{c} = t \hat{i} - t \hat{j} + \hat{k}, t \in R

be such that for

α, β, γ \in R, α \vec{a} + β \vec{b} + γ \vec{c} = \vec{0}

\Rightarrow α = β = γ = 0

. Then, the set of all values of

t

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 July 2022 Shift 1>Q 112

Let a vector

\vec{a}

has a magnitude

9

. Let a vector

\vec{b}

be such that for every

(x, y) R \times R - \{(0, 0)\}

, the vector

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

is perpendicular to the vector

(6 y \vec{a} - 18 x \vec{b})

. Then the value of

|\vec{a} \times \vec{b}|

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 July 2022 Shift 1>Q 113

Let

P (- 2, - 1, 1)

and

Q (\frac{56}{17}, \frac{43}{17}, \frac{111}{17})

be the vertices of the rhombus

P R Q S

. If the direction ratios of the diagonal

R S

are

α, - 1, β

, where both

α

and $\beta$ are integers of minimum absolute values, then

α^{2} + β^{2}

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 July 2022 Shift 2>Q 114

A plane

P

is parallel to two lines whose direction ratios are

- 2, 1, - 3

, and

- 1, 2, - 2

and it contains the point

(2, 2, - 2)

. Let

P

intersect the co-ordinate axes at the points

A, B, C

making the intercepts

α, β, γ

. If

V

is the volume of the tetrahedron

O A B C

, where

O

is the origin and

p = α + β + γ

, then the ordered pair

(V, p)

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 July 2022 Shift 2>Q 115

Let

S

be the set of all

a \in R

for which the angle between the vectors

\vec{u} = a (\log_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}

and

\vec{v} = (\log_{e} b) \hat{i} + 2 \hat{j} + 2 a (\log_{e} b) \hat{k}, (b > 1)

is acute. Then

S

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 29 July 2022 Shift 1>Q 116

Let

\vec{a} = 3 \hat{i} + \hat{j}

and

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

. Let

\vec{c}

be a vector satisfying

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

. If

\vec{b}

and

\vec{c}

are non-parallel, then the value of

λ

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 29 July 2022 Shift 1>Q 117

Let

\hat{a}

and

\hat{b}

be two unit vectors such that the angle between them is

\frac{π}{4}

. If

θ

is the angle between the vectors

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

and

(\hat{a} + 2 \hat{b} + 2 (\hat{a} \times \hat{b}))

then the value of

164 \cos^{2} θ

is equal to

Let a→=2i^-j^+5k^ and b→=αi^+βj^+2k^. If a→×b→×i^·k^=232, then b→×2j^ is equal to

Important Questions on Vector Algebra

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Let $\vec{a} = 2 \hat{i} - \hat{j} + 5 \hat{k}$ and $\vec{b} = α \hat{i} + β \hat{j} + 2 \hat{k}$ . If $((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k} = \frac{23}{2}$ , then $|\vec{b} \times 2 \hat{j}|$ is equal to