Before You Begin

Question #41621

Difficulty: 8/10 | 🟢 Free

The Question

A conservative vector field is given by $\vec{F}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j}$ . It is known that $M(x,y) = 2x e^{x^2 - y^2} + 3y$ and the scalar potential function $\phi(x,y)$ associated with $\vec{F}$ satisfies $\phi(0,y) = e^{-y^2}$ . Find the unit vector in the direction of $\vec{F}$ at the point $(1, 1)$ .

Options

A.
$\frac{5}{\sqrt{26}}\hat{i} + \frac{1}{\sqrt{26}}\hat{j}$
B.
$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}$
C.
$\frac{5}{\sqrt{29}}\hat{i} - \frac{2}{\sqrt{29}}\hat{j}$
D.
$\frac{5}{\sqrt{26}}\hat{i} - \frac{1}{\sqrt{26}}\hat{j}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Unit Vector Normalization|Exact Differential Recognition|Chain Rule Complexity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to reconstruct a conservative vector field from its partial components and a boundary condition, specifically targeting the chain rule during partial differentiation.

The trap lies in calculating $N(x,y) = \frac{\partial \phi}{\partial y}$ .

Students frequently miss the negative sign when applying the chain rule to the exponent $-y^2$ , incorrectly yielding a derivative of $2y e^{x^2 - y^2}$ .

This error leads to $N(1,1) = 5$ , resulting in the highly deceptive distractor vector $5\hat{i} + 5\hat{j}$ and its normalized form $\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}$ .

The clean magnitude of this incorrect vector creates false confidence.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Unit Vector Normalization
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free