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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Let $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{c} = 2 \hat{i} - 3 \hat{j} + 2 \hat{k}$ . Then the number of vectors $\vec{b}$ such that $\vec{b} \times \vec{c} = \vec{a}$ and $|\vec{b}| \in \{1, 2, \dots, 10\}$ is

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Important Questions on Vector Algebra

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JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 27 June 2022 Shift 2>Q 95

The shortest distance between the lines

\frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{- 1}

and

\frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3}

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 27 June 2022 Shift 2>Q 96

Let

\vec{a}

and

\vec{b}

be the vectors along the diagonal of a parallelogram having area

2 \sqrt{2}

. Let the angle between

\vec{a}

and

\vec{b}

be acute.

|\vec{a}| = 1

and

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

. If

\vec{c} = 2 \sqrt{2} (\vec{a} \times \vec{b}) - 2 \vec{b}

, then an angle between

\vec{b}

and

\vec{c}

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 June 2022 Shift 1>Q 97

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

and

\vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}

are coplanar vectors and

\vec{a} \cdot \vec{c} = 5, \vec{b} ⊥ \vec{c}

, then

122 (c_{1} + c_{2} + c_{3})

is equal to ______.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 June 2022 Shift 2>Q 98

Let

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

and

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

, where

α \in R

. If the area of the parallelogram whose adjacent sides are represented by the vectors

\vec{a}

and

\vec{b}

\sqrt{15 (α^{2} + 4)}

, then the value of

2 {|\vec{a}|}^{2} + (\vec{a} \cdot \vec{b}) {|\vec{b}|}^{2}

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 28 June 2022 Shift 2>Q 99

Let

\vec{a}

be a vector which is perpendicular to the vector

3 \hat{i} + \frac{1}{2} \hat{j} + 2 \hat{k}

. If

\vec{a} \times (2 \hat{i} + \hat{k}) = 2 \hat{i} - 13 \hat{j} - 4 \hat{k}

, then the projection of the vector

\vec{a}

on the vector

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

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 29 June 2022 Shift 1>Q 100

Let

\vec{a} = α \hat{i} + 3 \hat{j} - \hat{k}, \vec{b} = 3 \hat{i} - β \hat{j} + 4 \hat{k}

and

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

where

α, β \in R

be three vectors. If the projection of

\vec{a}

\vec{c}

\frac{10}{3}

and

\vec{b} \times \vec{c} = - 6 \hat{i} + 10 \hat{j} + 7 \hat{k}

, then the value of

α + β

equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 29 June 2022 Shift 2>Q 101

Let $A, B, C$ be three points whose position vectors respectively are:

$\vec{a} = \hat{i} + 4 \hat{j} + 3 \hat{k}$

$\vec{b} = 2 \hat{i} + α \hat{j} + 4 \hat{k}, α \in R$

$\vec{c} = 3 \hat{i} - 2 \hat{j} + 5 \hat{k}$

If $α$ is the smallest positive integer for which $\vec{a}, \vec{b}, \vec{c}$ are non-collinear, then the length of the median of $△ A B C$ through $A$ is:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 19 - Vector Algebra>JEE Main - 29 June 2022 Shift 2>Q 102

Let

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

and

\vec{c}

be a vector such that

\vec{a} \times (\vec{b} + \vec{c}) = \vec{0}

, then the value of

3 (\vec{c} . \vec{a})

is equal to _______.

Let a→=i^+j^-k^ and c→=2i^-3j^+2k^. Then the number of vectors b→ such that b→×c→=a→ and b→∈1,2,…,10 is

Important Questions on Vector Algebra

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Let $\vec{a} = \hat{i} + \hat{j} - \hat{k}$ and $\vec{c} = 2 \hat{i} - 3 \hat{j} + 2 \hat{k}$ . Then the number of vectors $\vec{b}$ such that $\vec{b} \times \vec{c} = \vec{a}$ and $|\vec{b}| \in \{1, 2, \dots, 10\}$ is