Kerala KEAM 5 June 2024 Solved Paper

The volume of the parallelepiped formed by three vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ is given by the scalar triple product:$$V = \left | \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \right | .$$Step 1: Compute the cross product $\overrightarrow{b} \times \overrightarrow{c}$: $$\overrightarrow{b} = 3i + j - k, \quad \overrightarrow{c} = 5i + 2j - 7k.$$The cross product is: $$\overrightarrow{b} \times \overrightarrow{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} 3 & 1 & -1 5 & 2 & -7 \end{vmatrix}.$$Using the determinant formula: $$= \hat{i} \begin{vmatrix} 1 & -1 2 & -7 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & -1 5 & -7 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 1 5 & 2 \end{vmatrix}.$$This simplifies to: $$= \hat{i} \left( 1(-7) - (-1)(2) \right) - \hat{j} \left( 3(-7) - (-1)(5) \right) + \hat{k} \left( 3(2) - 1(5) \right)$$ $$= \hat{i} \left( -7 + 2 \right) - \hat{j} \left( -21 + 5 \right) + \hat{k} \left( 6 - 5 \right)$$ $$= -5\hat{i} + 16\hat{j} + \hat{k}.$$Thus, $$\overrightarrow{b} \times \overrightarrow{c} = -5\hat{i} + 16\hat{j} + \hat{k}.$$Step 2: Compute the dot product $\overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c})$: $$\overrightarrow{a} = i - j + k.$$Now, take the dot product: $$\overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) = (i - j + k) \cdot (-5\hat{i} + 16\hat{j} + \hat{k}).$$This gives: $$= 1(-5) + (-1)(16) + 1(1) = -5 - 16 + 1 = -20.$$Step 3: The volume is the absolute value of this result: $$V = \left | -20 \right | = 20.$$Thus, the volume of the parallelepiped is 20 cubic units.Therefore, the correct answer is option (B).Quick Tip: The volume of a parallelepiped can be found using the scalar triple product $\left | \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \right |$. Remember to compute the cross product first and then the dot product.