Home >
Master Archives >
QB_EM_VC_001
Question Bank • QB_EM_VC_001
NAT • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Vector differentiation
Question Statement:
The directional derivative of the scalar field $\phi(x, y, z) = x^2 y z + 4 x z^2$ at the point $P(1, -2, -1)$ in the direction of the vector $\vec{a} = 2\hat{i} - \hat{j} - 2\hat{k}$ is ________ (round off to 2 decimal places).
Step-by-Step Solution & Explanation
1. Compute the gradient $\nabla \phi$:
$\nabla \phi = \left(2xyz + 4z^2\right)\hat{i} + \left(x^2 z\right)\hat{j} + \left(x^2 y + 8xz\right)\hat{k}$
At $P(1, -2, -1)$:
$\frac{\partial \phi}{\partial x} = 2(1)(-2)(-1) + 4(-1)^2 = 4 + 4 = 8$
$\frac{\partial \phi}{\partial y} = (1)^2(-1) = -1$
$\frac{\partial \phi}{\partial z} = (1)^2(-2) + 8(1)(-1) = -2 - 8 = -10$
$\nabla \phi = 8\hat{i} - \hat{j} - 10\hat{k}$
2. Find the unit direction vector $\hat{a}$:
$\hat{a} = \frac{2\hat{i} - \hat{j} - 2\hat{k}}{\sqrt{2^2 + (-1)^2 + (-2)^2}} = \frac{2\hat{i} - \hat{j} - 2\hat{k}}{3}$
3. Directional derivative:
$\nabla \phi \cdot \hat{a} = \frac{8(2) + (-1)(-1) + (-10)(-2)}{3} = \frac{16 + 1 + 20}{3} = \frac{37}{3} \approx 12.33$
Practice In Authentic CBT Exam Interface
Test your skills under authentic timed conditions with live scoring and AIR percentile ranking.
Launch CBT Mock Test