If a=2i+j, and b=j-k, then which of the following statements is/are TRUE ? A., a+b, =3 B. Unit vector along a+b is 2i+2j-k C. Direction ratio of a+b are 2, 2, -1 D. unit vector perpendicular to a and b is -1/3i+2/3j+2/3k Choose the correct answer from the options given below: Option 1: A and B only Option 2: A and C only Option 3: A, C and D only Option 4: B, C and D only

Answer: A, C and D only. Solution: a+ b=2 i+2 j- k, so its magnitude is 3 (A true) and its direction ratios are 2,2,-1 (C true). The unit vector is 13(2 i+2 j- k), so B is false. Also ax b=- i+2 j+2 k with magnitude 3, giving the unit normal in D, which is true.

Algebra

Vector Algebra

Medium

core

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

A. $|\vec{a}+\vec{b}|=3$

B. Unit vector along $\vec{a}+\vec{b}$ is $2\hat{i}+2\hat{j}-\hat{k}$

C. Direction ratio of $\vec{a}+\vec{b}$ are 2, 2, -1

D. unit vector perpendicular to $\vec{a}$ and $\vec{b}$ is $-\frac{1}{3}\hat{i}+\frac{2}{3}\hat{j}+\frac{2}{3}\hat{k}$

Choose the correct answer from the options given below:

A and B only

A and C only

A, C and D only

B, C and D only

✅ Correct Option: 3

$\vec a+\vec b=2\hat i+2\hat j-\hat k$ , so its magnitude is 3 (A true) and its direction ratios are $2,2,-1$ (C true). The unit vector is $\frac13(2\hat i+2\hat j-\hat k)$ , so B is false. Also $\vec a\times\vec b=-\hat i+2\hat j+2\hat k$ with magnitude 3, giving the unit normal in D, which is true.

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: 12 May Shift 2

If $\vec{a}.\vec{b} = \vec{a}.\vec{c}$ and $\vec{a}\times\vec{b} = \vec{a}\times\vec{c}$ , $\vec{a}\neq\vec{0}$ , then

2026: 21 May Shift 1

If $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$ , then which of the following statements are TRUE?

(A) unit vector along $\vec{a}$ is $\frac{1}{3}(2\hat{i}+2\hat{j}-\hat{k})$

(B) $|3\vec{a}| = 9$

(D) Direction Ratios of $\vec{a}$ are 2,-2,1

Choose the correct answer from the options given below:

2025: 27 May Shift 1

Match List-I with List-II

List-I	List-II
(A) If vector $\vec{a}$ and $\vec{b}$ are such that $\vec{a} = \lambda \vec{b}$ and $\vert \vec{a}\vert = \vert \vec{b}\vert$ , then	(I) $\vec{a}$ and $\vec{b}$ are orthogonal
(B) Projection vector of $\vec{a}$ on $\vec{b}$	(II) $[0, 12]$
(C) $\vec{a}$ and $\vec{b}$ are non-zero vectors such that $\vert \vec{a} + \vec{b}\vert = \vert \vec{a} - \vec{b}\vert$ , then	(III) $\vec{a} = \pm \vec{b}$
(D) If $\vert \vec{a}\vert = 4, -3 \le \lambda \le 2$ , then the range of $\vert \lambda \vec{a}\vert$	(IV) $(\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2}) \vec{b}$

Choose the correct answer from the options given below:

2023: 23 May Shift 3

The unit vector in the direction of $\vec{a} + \vec{b}$ if $\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}$ & $\vec{b} = -\hat{i} + \hat{j} + -\hat{k}$ is :

2025: 21 May Shift 1

If $\vec{a}$ , $\vec{b}$ and $\sqrt{3}\vec{a} - \vec{b}$ are three unit vectors, then the angle between $\vec{a}$ and $\vec{b}$ is:

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