Which of the following statements are true? (A) If r = xi + yj + zk, then x, y, z are called direction ratios of r. (B) For any two vectors a and b, a + b = b + a (C) a b if and only if a x b = 0 (D) Projection of b on a is (a * b)/(, a, ^2) Choose the correct answer from the options given below: Option 1: (B) and (C) only Option 2: (B), (C) and (D) only Option 3: (C) and (D) only Option 4: (A) and (B) only

Answer: (A) and (B) only. Solution: Solution Coming Soon

Algebra

Vector Algebra

Medium

core

Which of the following statements are true?

(A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ , then $x, y, z$ are called direction ratios of $\vec{r}$ .

(B) For any two vectors $\vec{a}$ and $\vec{b}$ , $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

(C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$

(D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$

Choose the correct answer from the options given below:

(B) and (C) only

(B), (C) and (D) only

(A) and (B) only

✅ Correct Option: 4

Solution Coming Soon

2022: 16 July Shift 2

If $|\vec{a}| = |\vec{b}| = |\vec{a} + \vec{b}| = 1$ , then $|\vec{a} - \vec{b}|$ is equal to:

2025: 16 May Shift 1

Match List-I with List-II

List-I	List-II
(A) Angle between î and -ĵ is	(I) $\frac{\pi}{6}$
(B) Angle between 2î + k̂ and 10î + 5k̂ is	(II) $\frac{\pi}{4}$
(C) Angle between î and î + ĵ is	(III) 2π
(D) Angle between √3ĵ - k̂ and ĵ is	(IV) $\frac{\pi}{2}$

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: 30 May Shift 3

If $|\vec{a}| = 3$ and $|\vec{b}| = 4$ , then a value of $\lambda$ for which $\vec{a} + \lambda \vec{b}$ and $\vec{a} - \lambda \vec{b}$ are perpendicular is :

2022: 30 Aug Shift 1

If $\vec{p} = \hat{i} + \hat{j} - 2\hat{k}$ and $\vec{q} = 2\hat{i} + \hat{j} - \hat{k}$ , then the area of parallelogram having diagonals $(\vec{p} + \vec{q})$ and $(\vec{p} - \vec{q})$ is

2025: 19 May Shift 1

If $|\vec{a}| = 1$ , $|\vec{b}| = 2$ , $|2\vec{a}+\vec{b}| = 2\sqrt{3}$ then $|\vec{a}-\vec{b}|$ is:

Solution

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