Algebra

Vector Algebra

Medium

core

If A (3, 2), B (1, -1) and C (2, 1) are three vertices of a parallelograms ABCD, then its area (in sq.units) is equal to

$\frac{1}{2}$

$\frac{5}{2}$

✅ Correct Option: 1

Given three vertices of parallelogram ABCD:

A(3, 2)

B(1, -1)

C(2, 1)

The area of a parallelogram can be found using any three consecutive vertices by calculating the cross product of two adjacent sides.

Vector AB represents the side from A to B:

Vector AB = B - A

Vector AB = (1 - 3, -1 - 2)

Vector AB = (-2, -3)

Vector AC represents the side from A to C:

Vector AC = C - A

Vector AC = (2 - 3, 1 - 2)

Vector AC = (-1, -1)

For a parallelogram with adjacent sides represented by vectors (x₁, y₁) and (x₂, y₂):

Area = $|x₁ \times y₂ - x₂ \times y₁|$

Substituting AB = (-2, -3) and AC = (-1, -1):

Area = $|(-2) \times (-1) - (-1) \times (-3)|$

Area = $|2 - 3|$

Area = $|-1|$

Area = $1$

Therefore, the area of parallelogram ABCD is 1 square unit.

2026: 19 May Shift 1

Let $\vec{a} = (\hat{i}+4\hat{j}+2\hat{k})$ , $\vec{b} = (3\hat{i}-2\hat{j}+7\hat{k})$ , $\vec{c} = (2\hat{i}-\hat{j}+4\hat{k})$ . A vector $\vec{d}$ , which is perpendicular to both $\vec{a}$ and $\vec{b}$ , such that $\vec{c}.\vec{d} = 18$ , is

2026: 18 May Shift 2

The vector $\{(a^2+a-2)\hat{i}+(a^2-4)\hat{j}-(a^2-5a+6)\hat{k}\}$ makes acute angles with coordinate axes, then values of $a$ lies in :

2025: 3 June Shift 1

Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$ , $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 4\hat{k}$ . A vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ , and $\vec{c} \cdot \vec{d} = 14$ , is:

2025: 13 May Shift 1

The vector equation of the line passing through points $A(3,4,-7)$ and $B(1,-1,6)$ 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:

2025: 27 May Shift 1

The projection of the vector $5\hat{i} + \hat{j} - 3\hat{k}$ on the vector $\hat{i} + 2\hat{j} - 3\hat{k}$ is

If A (3, 2), B (1, -1) and C (2, 1) are three vertices of a parallelograms ABCD, then its area (in sq.units) is equal to

Solution

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