Question

The vector \( \mathbf{r} = 3\hat{i} + 4\hat{k} \) can be written as the sum of a vector \( \mathbf{v} \), parallel to \( \hat{i} + \hat{k} \), and a vector \( \mathbf{u} \), perpendicular to \( \hat{i} + \hat{k} \). Then, the value of \( \mathbf{v} \) is

• A. \( \mathbf{v} = 3\hat{i} + 2\hat{k} \)
• B. \( \mathbf{v} = 4\hat{i} + \hat{k} \)
• C. \( \mathbf{v} = \hat{i} + 4\hat{k} \)
• D. \( \mathbf{v} = 3\hat{i} + \hat{k} \)

Hint:

To decompose a vector into components, use the projection formula to find the vector parallel to a given direction.

Answer 1

The Correct Option is A

Step 1: Decomposing the vector.
To find the component of the vector along \( \hat{i} + \hat{k} \), use the projection formula. The resulting vector will be \( 3\hat{i} + 2\hat{k} \).

Step 2: Conclusion.
The correct value of \( \mathbf{v} \) is \( 3\hat{i} + 2\hat{k} \), corresponding to option (1).