Question

A vector \( \vec{F}_1 \) is a unit vector along the positive direction of x-axis and \( \vec{F}_2 \) is of magnitude 4. If the vector product of \( \vec{F}_1 \) and \( \vec{F}_2 \) is zero, then \( \vec{F}_2 \) is

• A. \( 4\hat{j} \)
• B. \( 2\sqrt{3} \hat{i} - 2 \hat{j} \)
• C. \( 2 \hat{i} + 2\sqrt{3} \hat{j} \)
• D. \( -4\hat{i} \)

Hint:

When the vector product is zero, the vectors are parallel and must be in the same direction or opposite directions.

Answer 1

The Correct Option is C

Step 1: Understanding the vector product.
The vector product of two vectors \( \vec{A} \) and \( \vec{B} \) is zero when the vectors are parallel. That is, \( \vec{F}_1 \) and \( \vec{F}_2 \) must be parallel.
Step 2: Analyzing the given vectors.
The vector \( \vec{F}_1 \) is a unit vector along the positive x-axis, so \( \vec{F}_1 = \hat{i} \). The vector \( \vec{F}_2 \) must have components in both \( i \) and \( j \)-directions to be parallel to \( \hat{i} \), and its magnitude must be 4. The correct vector that satisfies this condition is: \[ \vec{F}_2 = 2 \hat{i} + 2\sqrt{3} \hat{j} \] Step 3: Conclusion.
Thus, \( \vec{F}_2 \) is \( 2 \hat{i} + 2\sqrt{3} \hat{j} \), corresponding to option (C).