Related Formula
Area of parallelogram = (1)/(2) | d₁ × d₂|$$\text{Area of parallelogram} = \frac{1}{2} |\overrightarrow{d_1} \times \overrightarrow{d_2}|$$
Core Logic
The diagonals of the parallelogram are AC$\overrightarrow{AC}$ and BD$\overrightarrow{BD}$.
First, calculate the diagonal vector AC$\overrightarrow{AC}$:
AC = Position vector of C - Position vector of A$$\overrightarrow{AC} = \text{Position vector of } C - \text{Position vector of } A$$
AC = (-3 - 2) i + (4 - 3) j + (-2 - 5) k = -5 i + 1 j - 7 k$$\overrightarrow{AC} = (-3 - 2)\hat{i} + (4 - 3)\hat{j} + (-2 - 5)\hat{k} = -5\hat{i} + 1\hat{j} - 7\hat{k}$$
Alternatively, taking CA = 5 i - j + 7 k$\overrightarrow{CA} = 5\hat{i} - \hat{j} + 7\hat{k}$. Let's use CA$\overrightarrow{CA}$ or AC$\overrightarrow{AC}$, the magnitude of the cross product will be the same.
The second diagonal is given:
BD = i + 2 j + 3 k$$\overrightarrow{BD} = \hat{i} + 2\hat{j} + 3\hat{k}$$
Step 1: Finding Cross Product
Area = (1)/(2) | AC × BD|$$\overrightarrow{Area} = \frac{1}{2} |\overrightarrow{AC} \times \overrightarrow{BD}|$$
AC × BD = vmatrix i & j & k -5 & 1 & -7 1 & 2 & 3 vmatrix$$\overrightarrow{AC} \times \overrightarrow{BD} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -5 & 1 & -7 \\ 1 & 2 & 3 \end{vmatrix}$$
= i(3 - (-14)) - j(-15 - (-7)) + k(-10 - 1)$$= \hat{i}(3 - (-14)) - \hat{j}(-15 - (-7)) + \hat{k}(-10 - 1)$$
= i(17) - j(-8) + k(-11)$$= \hat{i}(17) - \hat{j}(-8) + \hat{k}(-11)$$
= 17 i + 8 j - 11 k$$= 17\hat{i} + 8\hat{j} - 11\hat{k}$$
Step 2: Calculating Magnitude
Area = (1)/(2) |17 i + 8 j - 11 k|$$\text{Area} = \frac{1}{2} |17\hat{i} + 8\hat{j} - 11\hat{k}|$$
= (1)/(2) √(17² + 8² + (-11)²)$$= \frac{1}{2} \sqrt{17^2 + 8^2 + (-11)^2}$$
= (1)/(2) √(289 + 64 + 121)$$= \frac{1}{2} \sqrt{289 + 64 + 121}$$
= (1)/(2) √(474)$$= \frac{1}{2} \sqrt{474}$$
Pattern Recognition
When opposite vertices and one full diagonal vector are provided, immediately calculate the second diagonal vector via displacement and evaluate half the magnitude of their cross product.
Chapter Mix
Class 12 Maths: Vector Algebra
