Match List-I with List-II, List-I, List-II, (A) The vectors + j + 2k and i + + k are perpendicular if λ is equal to, (I) 1, (B) The vectors 3i + 6j - k and 2i + 4j - are collinear if λ is equal to, (II) -1, (C) The number of vectors of unit-length which are perpendicular to both the vectors a = i + j + 2k and b = 3i - j + 5k is, (III) 2/3, (D) If a = 1 and a + b = 0, then b is equal to, (IV) 2, Choose the correct answer from the options given below: Option 1: (A) - (II), (B) - (III), (C) - (IV), (D) - (I) Option 2: (A) - (I), (B) - (IV), (C) - (II), (D) - (III) Option 3: (A) - (III), (B) - (IV), (C) - (II), (D) - (I) Option 4: (A) - (IV), (B) - (III), (C) - (II), (D) - (I)

Answer: (A) - (II), (B) - (III), (C) - (IV), (D) - (I). Solution: (A) Two vectors are perpendicular when their dot product = 0 Given vectors: + j + 2k and i + + k ()(1) + (1)() + (2)(1) = 0 2 + 2 = 0 = -1 (A) arrow (II) (B) Two vectors are collinear when the ratios of their corresponding components are equal. Given vectors: 3i + 6j - k and 2i + 4j - 3/2 = 6/4 = (-1)/(-) 3/2 = 1/() = 2/3 (B) arrow (III) (C) The cross product a x b gives a vector perpendicular to both a and b. Since we can also take -(a x b), there are always exactly 2 unit vectors perpendicular to both given vectors (one in each direction along the perpendicular axis). (C) arrow (IV) (D) From a + b = 0: b = -a, b, =, -a, =, a, = 1 Magnitude is always positive, so the negative sign does not affect it. (D) arrow (I) (A) arrow (II), (B) arrow (III), (C) arrow (IV), (D) arrow (I)

Algebra

Vector Algebra

Medium

core

Match List-I with List-II

List-I List-II
(A) The vectors $\lambda\hat{i}$ $+ \hat{j} + 2\hat{k}$ and $\vec{i} + \lambda\hat{j} + \hat{k}$ are perpendicular if λ is equal to (I) 1
(B) The vectors $3\hat{i} + 6\hat{j} - \hat{k}$ and $2\hat{i} + 4\hat{j} - \lambda\hat{k}$ are collinear if λ is equal to (II) -1
(C) The number of vectors of unit-length which are perpendicular to both the vectors $\vec{a}$ = $\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b}$ = $3\hat{i} - \hat{j} + 5\hat{k}$ is (III) 2/3
(D) If $\lvert \vec{a} \rvert = 1$ and $\vec{a} + \vec{b} = \vec{0}$ , then $\lvert \vec{b} \rvert$ is equal to (IV) 2

Choose the correct answer from the options given below:

List-I	List-II
(A) The vectors $\lambda\hat{i}$ $+ \hat{j} + 2\hat{k}$ and $\vec{i} + \lambda\hat{j} + \hat{k}$ are perpendicular if λ is equal to	(I) 1
(B) The vectors $3\hat{i} + 6\hat{j} - \hat{k}$ and $2\hat{i} + 4\hat{j} - \lambda\hat{k}$ are collinear if λ is equal to	(II) -1
(C) The number of vectors of unit-length which are perpendicular to both the vectors $\vec{a}$ = $\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b}$ = $3\hat{i} - \hat{j} + 5\hat{k}$ is	(III) 2/3
(D) If $\lvert \vec{a} \rvert = 1$ and $\vec{a} + \vec{b} = \vec{0}$ , then $\lvert \vec{b} \rvert$ is equal to	(IV) 2

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

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

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

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

✅ Correct Option: 1

(A) Two vectors are perpendicular when their dot product $= 0$

Given vectors: $\lambda\hat{i} + \hat{j} + 2\hat{k}$ and $\hat{i} + \lambda\hat{j} + \hat{k}$

$(\lambda)(1) + (1)(\lambda) + (2)(1) = 0$

$2\lambda + 2 = 0$

$\lambda = -1$

(A) $\rightarrow$ (II)

(B) Two vectors are collinear when the ratios of their corresponding components are equal.

Given vectors: $3\hat{i} + 6\hat{j} - \hat{k}$ and $2\hat{i} + 4\hat{j} - \lambda\hat{k}$

$\dfrac{3}{2} = \dfrac{6}{4} = \dfrac{-1}{-\lambda}$

$\dfrac{3}{2} = \dfrac{1}{\lambda}$

$\lambda = \dfrac{2}{3}$

(B) $\rightarrow$ (III)

(C) The cross product $\vec{a} \times \vec{b}$ gives a vector perpendicular to both $\vec{a}$ and $\vec{b}$ . Since we can also take $-(\vec{a} \times \vec{b})$ , there are always exactly $2$ unit vectors perpendicular to both given vectors (one in each direction along the perpendicular axis).

(D) From $\vec{a} + \vec{b} = \vec{0}$ :

$\vec{b} = -\vec{a}$

$|\vec{b}| = |-\vec{a}| = |\vec{a}| = 1$

Magnitude is always positive, so the negative sign does not affect it.

(D) $\rightarrow$ (I)

(A) $\rightarrow$ (II), (B) $\rightarrow$ (III), (C) $\rightarrow$ (IV), (D) $\rightarrow$ (I)

2026: 14 May Shift 2

Which of the following statement are true?

A. The angle between the vectors $2\hat i+\hat j+3\hat k$ and $3\hat i-2\hat k$ is $\frac{\pi}{2}$

B. If $\vec a$ and $\vec b$ are two non-zero vectors, then projection of $\vec b$ on $\vec a$ is $\frac{\vec a\cdot\vec b}{|\vec a|}$

C. The unit vector normal to both vectors $\vec a=\hat i-\hat j-\hat k$ and $\vec b=\hat i+\hat j+\hat k$ is $\frac{1}{\sqrt2}(-\hat j+\hat k)$

D. The area of triangle formed by adjacent sides represented by vectors $\vec a=3\hat i+4\hat j$ and $\vec b=-5\hat i+7\hat j$ is 41 sq. units

Choose the correct answer from the options given below:

2025: 3 June Shift 1

Let $\vec{a} = 3\hat{i} + \hat{j} - 4\hat{k}$ and $\vec{b} = 6\hat{i} + 5\hat{j} - 2\hat{k}$ be two vectors. Then a vector perpendicular to $\vec{a}$ and $\vec{b}$ with magnitude 3 units is

2025: 3 June Shift 1

The projection of the vector $2\hat{i} - \hat{j} + 3\hat{k}$ on the vector $3\hat{i} + 2\hat{j} + 6\hat{k}$ 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:

2025: 19 May Shift 1

If $\vec{a} = 3\hat{i} - 6\vec{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - 4\vec{j} + \lambda\hat{k}$ are such that $\vec{a} \parallel \vec{b}$ , then $3\lambda + 2 =$

2026: 22 May Shift 1

Vector projection of $7\hat i+\hat j+4\hat k$ on the vector $2\hat i+6\hat j-3\hat k$ is

Solution

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