Question

If \[ \vec{a} = \hat{i} + 3\hat{j} + 13\hat{k}, \quad \vec{b} = 2\hat{i} - 4\hat{j} + 3\hat{k} \] are two vectors, then the component vector of \(\vec{a}\) perpendicular to \(\vec{b}\) is

• A. \(\hat{i} - \hat{j} - 2\hat{k}\)
• B. \(3\hat{i} + 3\hat{j} + 2\hat{k}\)
• C. \(\mathbf{-\hat{i} + 7\hat{j} + 10\hat{k}}\)
• D. \(4\hat{i} + 5\hat{j} + 4\hat{k}\)

Hint:

To find the vector component perpendicular to another, subtract the projection from the original vector.

Answer 1

The Correct Option is C

Component of \(\vec{a}\) perpendicular to \(\vec{b}\) is given by: \[ \vec{a}_{\perp} = \vec{a} - \text{proj}_{\vec{b}}\vec{a} = \vec{a} - \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b} \] \[ \vec{a} \cdot \vec{b} = (1)(2) + (3)(-4) + (13)(3) = 2 - 12 + 39 = 29 \quad |\vec{b}|^2 = 2^2 + (-4)^2 + 3^2 = 4 + 16 + 9 = 29 \] \[ \Rightarrow \text{proj}_{\vec{b}}\vec{a} = \left(\frac{29}{29}\right)\vec{b} = \vec{b} \Rightarrow \vec{a}_{\perp} = \vec{a} - \vec{b} = (\hat{i} + 3\hat{j} + 13\hat{k}) - (2\hat{i} - 4\hat{j} + 3\hat{k}) = -\hat{i} + 7\hat{j} + 10\hat{k} \]