For any vector a, the value of, a x i, ^2 +, a x j, ^2 +, a x k, ^2 is equal to: Option 1: 2, a, ^2 Option 2: 3, a, ^2 Option 3: 4, a, ^2 Option 4: a, ^2

Answer: 2, a, ^2. Solution: Let a = a_1i + a_2j + a_3k For a x i: a x i = (a_1i + a_2j + a_3k) x i Using i x i = 0, j x i = -k, and k x i = j: a x i = a_3j - a_2k, a x i, ^2 = a_3^2 + a_2^2 For a x j: a x j = (a_1i + a_2j + a_3k) x j Using i x j = k, j x j = 0, and k x j = -i: a x j = -a_3i + a_1k, a x j, ^2 = a_3^2 + a_1^2 For a x k: a x k = (a_1i + a_2j + a_3k) x k Using i x k = -j, j x k = i, and k x k = 0: a x k = a_2i - a_1j, a x k, ^2 = a_2^2 + a_1^2, a x i, ^2 +, a x j, ^2 +, a x k, ^2 = (a_2^2 + a_3^2) + (a_1^2 + a_3^2) + (a_1^2 + a_2^2) = 2a_1^2 + 2a_2^2 + 2a_3^2 = 2(a_1^2 + a_2^2 + a_3^2) = 2, a, ^2

Algebra

Vector Algebra

Medium

core

For any vector $\vec{a}$ , the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to:

$2|\vec{a}|^2$

$3|\vec{a}|^2$

$4|\vec{a}|^2$

$|\vec{a}|^2$

✅ Correct Option: 1

Let $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$

For $\vec{a} \times \hat{i}$ :

$\vec{a} \times \hat{i} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \times \hat{i}$

Using $\hat{i} \times \hat{i} = 0$ , $\hat{j} \times \hat{i} = -\hat{k}$ , and $\hat{k} \times \hat{i} = \hat{j}$ :

$\vec{a} \times \hat{i} = a_3\hat{j} - a_2\hat{k}$

$|\vec{a} \times \hat{i}|^2 = a_3^2 + a_2^2$

For $\vec{a} \times \hat{j}$ :

$\vec{a} \times \hat{j} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \times \hat{j}$

Using $\hat{i} \times \hat{j} = \hat{k}$ , $\hat{j} \times \hat{j} = 0$ , and $\hat{k} \times \hat{j} = -\hat{i}$ :

$\vec{a} \times \hat{j} = -a_3\hat{i} + a_1\hat{k}$

$|\vec{a} \times \hat{j}|^2 = a_3^2 + a_1^2$

For $\vec{a} \times \hat{k}$ :

$\vec{a} \times \hat{k} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \times \hat{k}$

Using $\hat{i} \times \hat{k} = -\hat{j}$ , $\hat{j} \times \hat{k} = \hat{i}$ , and $\hat{k} \times \hat{k} = 0$ :

$\vec{a} \times \hat{k} = a_2\hat{i} - a_1\hat{j}$

$|\vec{a} \times \hat{k}|^2 = a_2^2 + a_1^2$

$|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$

$= (a_2^2 + a_3^2) + (a_1^2 + a_3^2) + (a_1^2 + a_2^2)$

$= 2a_1^2 + 2a_2^2 + 2a_3^2$

$= 2(a_1^2 + a_2^2 + a_3^2)$

$= 2|\vec{a}|^2$

2026: 21 May Shift 2

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}| = 2, |\vec{b}| = 5, |\vec{c}| = 3$ and $\vec{a} + 2\vec{b} - 3\vec{c} = \vec{0}$ , then angle between $\vec{b}$ and $\vec{c}$ is equal to:

2025: 27 May Shift 1

Which of the following statements are true?

(A) The vector joining the points P(2, 3, 0) and Q(-1,-2,-4) directed from P to Q is $\vec{PQ} = -3\hat{i} - 5\hat{j} - 4\hat{k}$

(B) Projection of a vector $\vec{a}$ on other vector $\vec{b}$ is $\frac{\vec{a}.\vec{b}}{|\vec{a}|}$

(C) If $\vec{a} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}$ then $\vec{a} + \vec{b} = -\hat{i} + 2\hat{j} + 6\hat{k}$

(D) If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ then $\cos \theta = \frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$

Choose the correct answer from the options given below:

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:

2022: 16 July Shift 2

If $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ , $\vec{b} = 2\hat{i} + \hat{j} - 3\hat{k}$ , $\vec{c} = 2\hat{i} - \hat{j} + 7\hat{k}$ and $\vec{a} \times (\vec{b} \times \vec{c}) = \lambda \vec{b} + \mu \vec{c}$

(When $\lambda$ , $\mu$ are scalars), then the value of $\lambda + \mu$ is:

2026: 15 May Shift 1

For the vector $\vec a = \dfrac{1}{3}(2\hat i - 2\hat j + \hat k)$ , which of the following statements is/are TRUE ?

A. vector $\vec a$ is a unit vector

B. vector $\vec a$ is making an angle $\dfrac{\pi}{3}$ with vector $2\hat i - 4\hat j + 3\hat k$

C. vector $\vec a$ is parallel to vector $-\hat i + \hat j - \dfrac{1}{2}\hat k$

D. vector $\vec a$ is perpendicular to vector $3\hat i + 2\hat j - 2\hat k$

Choose the correct answer from the options given below:

2026: 26 May Shift 2

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

For any vector a⃗\vec{a}a, the value of ∣a⃗×i^∣2+∣a⃗×j^∣2+∣a⃗×k^∣2|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2∣a×i^∣2+∣a×j^​∣2+∣a×k^∣2 is equal to:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

For any vector $\vec{a}$ , the value of $|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is equal to: