Question

The orthogonal projection vector of \( \bar{a} = 2\bar{i} + 3\bar{j} + 3\bar{k} \) on \( \bar{b} = \bar{i} - 2\bar{j} + \bar{k} \) is:

• A. \( -\frac{1}{6} (2\bar{i} + 3\bar{j} + 3\bar{k}) \)
• B. \( \frac{1}{6} (-\bar{i} + 2\bar{j} - \bar{k}) \)
• C. \( \bar{i} - 2\bar{j} + \bar{k} \)
• D. \( -\bar{i} + 2\bar{j} - \bar{k} \)

Hint:

For vector projections, use the formula \( \frac{\bar{a} \cdot \bar{b}}{\bar{b} \cdot \bar{b}} \bar{b} \).

Answer 1

The Correct Option is B

Step 1: Formula for vector projection \[ \text{Proj}_{\bar{b}} \bar{a} = \frac{\bar{a} \cdot \bar{b}}{\bar{b} \cdot \bar{b}} \bar{b}. \] Step 2: Compute dot products \[ \bar{a} \cdot \bar{b} = (2)(1) + (3)(-2) + (3)(1) = 2 - 6 + 3 = -1. \] \[ \bar{b} \cdot \bar{b} = (1)^2 + (-2)^2 + (1)^2 = 1 + 4 + 1 = 6. \] Step 3: Compute projection \[ \text{Proj}_{\bar{b}} \bar{a} = \frac{-1}{6} (1\bar{i} - 2\bar{j} + 1\bar{k}) = \frac{1}{6} (-\bar{i} + 2\bar{j} - \bar{k}). \]