If a = 5i - j - 3k & b = i - 3j + 5k the angle between a + b and a - b is : Option 1: 60^ Option 2: 90^ Option 3: 120^ Option 4: 30^

Answer: 90^. Solution: a, ^2 = 25 + 1 + 9 = 35 and, b, ^2 = 1 + 9 + 25 = 35. Since the magnitudes are equal, (a+b) * (a-b) =, a, ^2 -, b, ^2 = 0. Hence the angle between them is 90 degrees.

Algebra

Vector Algebra

Easy

core

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ & $\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}$ the angle between $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is :

60 $^\circ$

90 $^\circ$

120 $^\circ$

30 $^\circ$

✅ Correct Option: 2

$|\vec{a}|^2 = 25 + 1 + 9 = 35$ and $|\vec{b}|^2 = 1 + 9 + 25 = 35$ . Since the magnitudes are equal, $(\vec{a}+\vec{b}) \cdot (\vec{a}-\vec{b}) = |\vec{a}|^2 - |\vec{b}|^2 = 0$ . Hence the angle between them is 90 degrees.

2025: 30 May Shift 2

A unit vector perpendicular to the vectors $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ is

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Area (in sq. units) of the triangle with vertices A (1,1,1) B (1,2,3) and C (2,3,1) is:

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If $\hat{a}, \hat{b}$ and $\hat{c}$ are three unit vectors and $\hat{a} + \hat{b} + \hat{c} = \vec{0}$ , then the angle between $\hat{a}$ and $(-\hat{b})$ is

2025: 14 May Shift 1

If $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors along co-ordinates axes OX, OY and OZ respectively, then which of the following is/are true?

(A) $\hat{i} \times \hat{i} = \vec{0}$

(B) $\hat{i} \times \hat{k} = \hat{j}$

(D) $\hat{i} \cdot \hat{j} = 0$

Choose the correct answer from the options given below:

2025: 3 June Shift 1

Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$ , $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 4\hat{k}$ . A vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ , and $\vec{c} \cdot \vec{d} = 14$ , is:

2022: 7 Aug Shift 2

Match List - I with List - II.

	List - I (Two given vector)		List - II (Projection of $\vec{a}$ on $\vec{b}$ )
(A)	$\vec{a} = \hat{i} - \hat{j}$ , $\vec{b} = \hat{i} + \hat{j}$	(I)	$\frac{2}{\sqrt{5}}$
(B)	$\vec{a} = \hat{i} + \hat{j}$ , $\vec{b} = 2\hat{i} - \hat{k}$	(II)	0
(C)	$\vec{a} = \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} + \hat{k}$	(III)	$\sqrt{2}$
(D)	$\vec{a} = 2\hat{i} + 3\hat{j}$ , $\vec{b} = \hat{i} - \hat{k}$	(IV)	$\frac{1}{\sqrt{2}}$

Choose the correct answer from the options given below :

If a⃗=5i^−j^−3k^\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}a=5i^−j^​−3k^ & b⃗=i^−3j^+5k^\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}b=i^−3j^​+5k^ the angle between a⃗+b⃗\vec{a} + \vec{b}a+b and a⃗−b⃗\vec{a} - \vec{b}a−b is :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ & $\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}$ the angle between $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is :