Question

Consider two vectors: \[ \mathbf{a} = 5 \hat{i} + 7 \hat{j} + 2 \hat{k}, \quad \mathbf{b} = 3 \hat{i} - \hat{j} + 6 \hat{k} \] Magnitude of the component of \( \mathbf{a} \) orthogonal to \( \mathbf{b} \) in the plane containing the vectors \( \mathbf{a} \) and \( \mathbf{b} \) is ________________ (round off to 2 decimal places).

Hint:

To find the component of one vector orthogonal to another, subtract the projection of the first vector onto the second from the first vector.

Answer 1

Correct Answer: 7.9

The component of \( \mathbf{a} \) orthogonal to \( \mathbf{b} \) is given by the formula: \[ \mathbf{a}_{\perp} = \mathbf{a} - \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b} \] First, calculate the dot products: \[ \mathbf{a} \cdot \mathbf{b} = (5)(3) + (7)(-1) + (2)(6) = 15 - 7 + 12 = 20 \] \[ \mathbf{b} \cdot \mathbf{b} = (3)^2 + (-1)^2 + (6)^2 = 9 + 1 + 36 = 46 \] Now, find the orthogonal component of \( \mathbf{a} \): \[ \mathbf{a}_{\perp} = \mathbf{a} - \left( \frac{20}{46} \right) \mathbf{b} = \left( 5 \hat{i} + 7 \hat{j} + 2 \hat{k} \right) - \left( \frac{20}{46} \right) \left( 3 \hat{i} - \hat{j} + 6 \hat{k} \right) \] Simplify: \[ \mathbf{a}_{\perp} = \left( 5 \hat{i} + 7 \hat{j} + 2 \hat{k} \right) - \left( 1.304 \hat{i} - 0.434 \hat{j} + 2.609 \hat{k} \right) \] \[ \mathbf{a}_{\perp} = \left( 3.696 \hat{i} + 7.434 \hat{j} - 0.609 \hat{k} \right) \] Now, find the magnitude of the orthogonal component: \[ |\mathbf{a}_{\perp}| = \sqrt{(3.696)^2 + (7.434)^2 + (-0.609)^2} = \sqrt{13.66 + 55.27 + 0.37} = \sqrt{69.3} \approx 8.33 \] Thus, the magnitude of the component of \( \mathbf{a} \) orthogonal to \( \mathbf{b} \) is \( \boxed{8.33} \).