Question

The component of a vector $ \vec{P} = 3\hat{i} + 4\hat{j} $ along the direction $ \hat{i} + 2\hat{j} $ is

• A. \( \dfrac{8}{\sqrt{5}} \)
• B. \( \dfrac{11}{\sqrt{5}} \)
• C. \( \dfrac{11}{2} \)
• D. \( \sqrt{10} \)

Hint:

To find the component of a vector in a given direction, take the dot product with the direction vector and divide by the magnitude of the direction vector.

Answer 1

The Correct Option is B

To find the component of \( \vec{P} \) in the direction of \( \vec{d} = \hat{i} + 2\hat{j} \), we use the formula: \[ \text{Component} = \frac{\vec{P} \cdot \vec{d}}{|\vec{d}|} \] First, compute the dot product: \[ \vec{P} \cdot \vec{d} = (3)(1) + (4)(2) = 3 + 8 = 11 \] Then the magnitude of \( \vec{d} \) is: \[ |\vec{d}| = \sqrt{1^2 + 2^2} = \sqrt{5} \] So the component is: \[ \frac{11}{\sqrt{5}} \]