Algebra

Vector Algebra

Medium

core

Which of the following statements are correct?

(A) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a triangle, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$

(B) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a parallelogram, then its area is $|\vec{a} \times \vec{b}|$

(C) $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \cos\theta$

(D) If $\vec{a}$ and $\vec{b}$ represent the 'diagonals' of a parallelogram, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$

Choose the correct answer from the options given below:

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

(A), (B) and (D) only

(A) and (D) only

✅ Correct Option: 3

For two sides of a triangle meeting at a point, they form an angle between them.

The magnitude of the cross product is:

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

where $\theta$ is the angle between the vectors.

Area of a triangle:

$= \frac{1}{2} \times \text{base} \times \text{height}$

$= \frac{1}{2}|\vec{a}||\vec{b}|\sin\theta$

$= \frac{1}{2}|\vec{a} \times \vec{b}|$

Statement (A) is correct.

Area of parallelogram:

$= \text{base} \times \text{height}$

$= |\vec{a}||\vec{b}|\sin\theta$

$= |\vec{a} \times \vec{b}|$

The parallelogram has twice the area of a triangle.

Statement (B) is correct.

The correct formula for cross product magnitude is:

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

The given formula uses $\cos\theta$ instead of $\sin\theta$ .

The dot product uses $\cos\theta$ , while the cross product uses $\sin\theta$ .

Statement (C) is incorrect.

When diagonals of a parallelogram are given as vectors $\vec{a}$ and $\vec{b}$ , the area is:

$\text{Area} = \frac{1}{2}|\vec{a} \times \vec{b}|$

Statement (D) is correct.

The correct statements are (A), (B), and (D).

2025: 19 May Shift 1

Match List-I with List-II

List-I	List-II
Mathematical Statement	Value
(A) $\hat{i} \cdot (\hat{j} \times \hat{k})$	(I) $-\hat{k}$
(B) $\hat{j} \cdot (\hat{i} \times \hat{k})$	(II) 1
(C) $\hat{i} \times (\hat{j} \times \hat{k})$	(III) -1
(D) $\hat{j} \times \hat{i}$	(IV) $\vec{0}$

Choose the correct answer from the options given below:

2022: 7 Aug Shift 2

The unit vector in the direction of the sum of vectors $\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ is :

2026: 21 May Shift 1

Let us consider vector $\vec{a} = (c\log_2 x)\hat{i} - 6\hat{j} + 3\hat{k}$ and $\vec{b} = (\log_2 x)\hat{i} + 2\hat{j} + (2c\log_2 x)\hat{k}$ , then which of following statements are TRUE?

(Where $x \in \mathbb{R}$ (set of real numbers))

(A) Vector $\vec{a}$ and $\vec{b}$ make obtuse angle for any $x \in (0,\infty)$ if $c \in (-1,0)$

(B) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in \mathbb{R}$ if $c > 0$

(C) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in (0,\infty)$ if $c \in \left(-\frac{4}{3}, -1\right]$

(D) Vectors $\vec{a}$ and $\vec{b}$ makes obtuse angle for $x \in \mathbb{R}$ if $c \in (1,2)$

Choose the correct answer from the options given below:

2025: 21 May Shift 2

If $\vec{a} = 2\hat{j} - \hat{k}$ , $\vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{c} = -\hat{i} + \hat{k}$ are three vectors, then the area (in sq. units) of the parallelogram whose diagonals are $(\vec{b} + \vec{c})$ and $(\vec{a} + \vec{c})$ is

2026: 29 May Shift 2

Let $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}|=2, |\vec{b}|=1, |\vec{c}|=3$ . If projection of $\vec{a}$ on $\vec{b}$ is equal to projection of $\vec{c}$ on $\vec{b}$ and $\vec{a}\perp\vec{c}$ , then value of $|2\vec{a}+3\vec{b}-2\vec{c}|$ is

2024: 16 May Shift 1

If $(\vec{a}-\vec{b}) \cdot(\vec{a}+\vec{b})=27$ and $|\vec{a}|=2|\vec{b}|$ , then $|\vec{b}|$ is :

Solution

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