If a = 2i + j and b = j - k, then Match List-I with List-II, LIST-I, LIST-II, A. Unit vector along a+b, I. 1/3(-i+2j+2k), B. Unit vector perpendicular to both a and b, II. 1, C. a+b, III. 1/3(2i+2j-k), D. a*b, IV. 3, Choose the correct answer from the options given below: Option 1: A-III, B-I, C-II, D-IV Option 2: A-I, B-III, C-IV, D-II Option 3: A-III, B-IV, C-I, D-II Option 4: A-III, B-I, C-IV, D-II

Answer: A-III, B-I, C-IV, D-II. Solution: a+b = 2i+2j-k has magnitude 3, so its unit vector is 1/3(2i+2j-k). Also axb = -i+2j+2k has magnitude 3, giving unit vector 1/3(-i+2j+2k), and a*b = 1.

Algebra

Vector Algebra

Medium

core

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:

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. $3$

A-III, B-I, C-II, D-IV

A-I, B-III, C-IV, D-II

A-III, B-IV, C-I, D-II

A-III, B-I, C-IV, D-II

✅ Correct Option: 4

$\vec{a}+\vec{b} = 2\hat{i}+2\hat{j}-\hat{k}$ has magnitude $3$ , so its unit vector is $\frac{1}{3}(2\hat{i}+2\hat{j}-\hat{k})$ .

Also $\vec{a}\times\vec{b} = -\hat{i}+2\hat{j}+2\hat{k}$ has magnitude $3$ , giving unit vector $\frac{1}{3}(-\hat{i}+2\hat{j}+2\hat{k})$ , and $\vec{a}\cdot\vec{b} = 1$ .

2025: 13 May Shift 1

The projection of the vector $\vec{a} = \hat{i} + 2\hat{j} - 3\hat{k}$ on the vector $2\hat{i} + 6\hat{j} + 3\hat{k}$ is

2026: 22 May Shift 1

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

2025: 13 May Shift 2

Match List-I with List-II

List-I	List-II
(A) Angle between $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$	(I) $\pi$
(B) Angle between $\hat{i} - \hat{j} + \hat{k}$ and $-\hat{i} + \hat{j} - \hat{k}$	(II) $\frac{3\pi}{4}$
(C) Angle between $\hat{i} + \hat{j}$ and $-\hat{i}$	(III) $\frac{\pi}{4}$
(D) Angle between $\hat{i} + \hat{k}$ and $\hat{k}$	(IV) $\frac{\pi}{2}$

Choose the correct answer from the options given below:

2022: 25 May Shift 1

Match List I with List II

LIST I	LIST II
A. The area of parallelogram determined by vectors $2\hat{i}$ and $3\hat{j}$	I. 2
B. The value of $(\hat{i} \times \hat{j}) \cdot \hat{k} + (\hat{j} \times \hat{k}) \cdot \hat{i}$	II. 4
C. The value of a for which the vectors $2\hat{i} - 3\hat{j} + 4\hat{k}$ and $a\hat{i} - 6\hat{j} + 8\hat{k}$ are collinear.	III. 0
D. The value of $\lambda$ for which the vectors $2\hat{i} + \hat{j} + \hat{k}$ and $2\hat{i} - 4\hat{j} + \lambda\hat{k}$ are perpendicular	IV. 6

Choose the correct answer from the options given below:

2026: 22 May Shift 2

A vector of magnitude 9, which is perpendicular to both the vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$ is:

2025: 13 May Shift 1

Match List-I with List-II

List-I	List-II
(A) Angle between $\vec{i} - \vec{j}$ and $\vec{j} + \vec{k}$	(I) 0
(B) Angle between $2\vec{j} - \vec{k}$ and $\vec{j} + 2\vec{k}$	(II) $\frac{2\pi}{3}$
(C) Angle between $\vec{i} + 2\vec{j}$ and $5\vec{i} + 10\vec{j}$	(III) $\frac{\pi}{6}$
(D) Angle between $\sqrt{3}\vec{i} + \vec{j}$ and $\vec{i}$	(IV) $\frac{\pi}{2}$

Choose the correct answer from the options given below:

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