If a = 2i - 3j + k and b = 2i + j - k, then which of the following statements is/are correct? (A) a and b are collinear (B) a and b are perpendicular (C) Angle between a and b is ()/4 (D), a + b, = 2√5 Choose the correct answer from the options given below: Option 1: (A) and (C) only Option 2: (B) only Option 3: (A), (C) and (D) only Option 4: (B) and (D) only

Answer: (B) and (D) only. Solution: Given a = 2i - 3j + k and b = 2i + j - k Statement (A): a and b are collinear Two vectors are collinear if a = kb for some scalar k. Comparing ratios of corresponding components: For i: 2/2 = 1 For j: (-3)/1 = -3 For k: 1/(-1) = -1 Since 1 ≠ -3 ≠ -1, the ratios are different. Statement (A) is FALSE. Statement (B): a and b are perpendicular Two vectors are perpendicular when their dot product equals zero. a * b = (2)(2) + (-3)(1) + (1)(-1) = 4 - 3 - 1 = 0 Since a * b = 0, the vectors are perpendicular. Statement (B) is TRUE. Statement (C): Angle between a and b is ()/4 From the previous result, a * b = 0, which means the vectors are perpendicular. The angle for perpendicular vectors is ()/2, not ()/4. Statement (C) is FALSE. Statement (D):, a + b, = 2√5 a + b = (2+2)i + (-3+1)j + (1-1)k = 4i - 2j + 0k, a + b, = √((4)^2 + (-2)^2 + (0)^2) = √(16 + 4 + 0) = √20 = √(4 x 5) = 2√5 Statement (D) is TRUE. Statements (B) and (D) only are correct.

Algebra

Vector Algebra

Medium

core

If $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$ , then which of the following statements is/are correct?

(A) $\vec{a}$ and $\vec{b}$ are collinear

(B) $\vec{a}$ and $\vec{b}$ are perpendicular

(C) Angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$

(D) $|\vec{a} + \vec{b}| = 2\sqrt{5}$

Choose the correct answer from the options given below:

(A) and (C) only

(B) only

(A), (C) and (D) only

(B) and (D) only

✅ Correct Option: 4

Given $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$

Statement (A): $\vec{a}$ and $\vec{b}$ are collinear

Two vectors are collinear if $\vec{a} = k\vec{b}$ for some scalar $k$ .

Comparing ratios of corresponding components:

For $\hat{i}$ : $\frac{2}{2} = 1$

For $\hat{j}$ : $\frac{-3}{1} = -3$

For $\hat{k}$ : $\frac{1}{-1} = -1$

Since $1 \neq -3 \neq -1$ , the ratios are different.

Statement (A) is FALSE.

Statement (B): $\vec{a}$ and $\vec{b}$ are perpendicular

Two vectors are perpendicular when their dot product equals zero.

$\vec{a} \cdot \vec{b} = (2)(2) + (-3)(1) + (1)(-1)$

$= 4 - 3 - 1$

$= 0$

Since $\vec{a} \cdot \vec{b} = 0$ , the vectors are perpendicular.

Statement (B) is TRUE.

Statement (C): Angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$

From the previous result, $\vec{a} \cdot \vec{b} = 0$ , which means the vectors are perpendicular.

The angle for perpendicular vectors is $\frac{\pi}{2}$ , not $\frac{\pi}{4}$ .

Statement (C) is FALSE.

Statement (D): $|\vec{a} + \vec{b}| = 2\sqrt{5}$

$\vec{a} + \vec{b} = (2+2)\hat{i} + (-3+1)\hat{j} + (1-1)\hat{k}$

$= 4\hat{i} - 2\hat{j} + 0\hat{k}$

$|\vec{a} + \vec{b}| = \sqrt{(4)^2 + (-2)^2 + (0)^2}$

$= \sqrt{16 + 4 + 0}$

$= \sqrt{20}$

$= \sqrt{4 \times 5}$

$= 2\sqrt{5}$

Statement (D) is TRUE.

Statements (B) and (D) only are correct.

2025: 21 May Shift 2

If $\vec{a} = 2\hat{j} - \hat{k}$ , $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{c} = -\hat{i} + \hat{k}$ are three vectors, then the area (in sq. units) of the parallelogram whose diagonals are $(\vec{b} + \vec{c})$ and $(\vec{a} + \vec{c})$ is

2026: 21 May Shift 1

Let us consider vector $\vec{a} = (c\log_2 x)\hat{i} - 6\hat{j} + 3\hat{k}$ and $\vec{b} = (\log_2 x)\hat{i} + 2\hat{j} + (2c\log_2 x)\hat{k}$ , then which of following statements are TRUE?

(Where $x \in \mathbb{R}$ (set of real numbers))

(A) Vector $\vec{a}$ and $\vec{b}$ make obtuse angle for any $x \in (0,\infty)$ if $c \in (-1,0)$

(B) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in \mathbb{R}$ if $c > 0$

(C) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in (0,\infty)$ if $c \in \left(-\frac{4}{3}, -1\right]$

(D) Vectors $\vec{a}$ and $\vec{b}$ makes obtuse angle for $x \in \mathbb{R}$ if $c \in (1,2)$

Choose the correct answer from the options given below:

2025: 21 May Shift 2

The area of triangle with vertices P, Q, R is given by (where $\vec{AB}$ = position vector of point B – position vector of point A)

2026: 14 May Shift 2

If $\vec a=\hat i+\hat j+2\hat k$ and $\vec b=2\hat i+\hat j+2\hat k$ , then the unit vector in the direction of $2\vec a-\vec b$ is equal to :

2025: 30 May Shift 1

Let $\vec{a} = 2\hat{i} - \hat{j}, \vec{b} =- 4\hat{j} + k\,\text{and}\,\vec{c} = \hat{i} + 2\hat{k}$ . If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 34$ , then $|\vec{d}|$ is equal to

2025: 15 May Shift 1

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ , $|\vec{a}| = 7$ , $|\vec{b}| = 3$ and $|\vec{c}| = 5$ , then angle between $\vec{b}$ and $\vec{c}$ is

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)