Algebra

Vector Algebra

Easy

core

If $\vec a, \vec b$ and $\vec c$ are three vectors, then meaningless expressions are

(A) $\vec a.(\vec b\times\vec c)$

(B) $\vec a.(\vec b.\vec c)$

(C) $\vec a\times(\vec b\times\vec c)$

(D) $\vec a\times(\vec b.\vec c)$

Choose the correct answer from the options given below:

(A) and (C) only

(A) and (B) only

(B) and (D) only

✅ Correct Option: 4

$\vec b\cdot\vec c$ is a scalar, so both $\vec a\cdot(\vec b\cdot\vec c)$ and $\vec a\times(\vec b\cdot\vec c)$ try to apply a vector operation between a vector and a scalar, which is meaningless.

(A) is the scalar triple product and (C) is the vector triple product, both perfectly well defined.

2026: 25 May Shift 1

If $\lvert\vec{a}\rvert = \lvert\vec{b}\rvert = \lvert\vec{a}+\vec{b}\rvert = 1$ , then

Match List-I with List-II

LIST-I	LIST-II
A. $\lvert\vec{a}-\vec{b}\rvert$	I. $\dfrac{2\pi}{3}$
B. Angle between $\vec{a}$ and $\vec{b}$	II. $-\dfrac{1}{2}$
C. Angle between $\vec{a}$ and $\vec{a}+\vec{b}$	III. $\sqrt3$
D. $\vec{a}.\vec{b}$	IV. $\dfrac{\pi}{3}$

Choose the correct answer from the options given below:

2026: 15 May Shift 1

If $|\vec a|=2, |\vec b|=7, \vec a \times \vec b = 3\hat i + 2\hat j + 6\hat k$ , then the angle between $\vec a$ and $\vec b$ is equal to

2025: 3 June Shift 2

Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$ . Then match List-I with List-II

List-I	List-II
(A) $\sin \theta$	(I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert \vert \vec{b}\vert }$
(B) $\cos \theta$	(II) $\vert \vec{a} \times \vec{b}\vert$
(C) Area of the parallelogram with adjacent sides represented by $\vec{a}$ and $\vec{b}$	(III) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert }$
(D) Projection of $\vec{a}$ on $\vec{b}$	(IV) $\dfrac{\vert \vec{a} \times \vec{b}\vert }{\vert \vec{a}\vert \vert \vec{b}\vert }$

Choose the correct answer from the options given below:

2026: 31 May Shift 2

If $\vec{a} = 2\hat{i} + \hat{j}$ and $\vec{b} = \hat{j} - \hat{k}$ , then Match List-I with List-II

LIST-I	LIST-II
A. Unit vector along $\vec{a}+\vec{b}$	I. $\dfrac{1}{3}\left(-\hat{i}+2\hat{j}+2\hat{k}\right)$
B. Unit vector perpendicular to both $\vec{a}$ and $\vec{b}$	II. $1$
C. $\left\vert \vec{a}+\vec{b} \right\vert$	III. $\dfrac{1}{3}\left(2\hat{i}+2\hat{j}-\hat{k}\right)$
D. $\vec{a}\cdot\vec{b}$	IV. $3$

Choose the correct answer from the options given below:

2026: 14 May Shift 2

If $\vec a, \vec b, \vec c$ are mutually perpendicular vectors of equal magnitude, the angle between $\vec a+\vec b+\vec c$ and $\vec a$ is

2022: 23 Aug Shift 1

If $-3 \leq k \leq 1$ and $|\vec{a}| = 2$ then $|k\vec{a}|$ is

Solution

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