Question

Given the vectors: \[ \mathbf{a} = i + 3j - k, \quad \mathbf{b} = 3i - j + 2k, \quad \mathbf{c} = i + 2j - 2k \] and the following information: \[ \frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{c}|} = \frac{10}{3} \] Find the value of \( \alpha + \beta \) and the projection of \( \mathbf{a} \) on \( \mathbf{c} \).

• A. \( \alpha + \beta = 30^\circ \), Projection of \( \mathbf{a} \) on \( \mathbf{c} = 5 \)
• B. \( \alpha + \beta = 45^\circ \), Projection of \( \mathbf{a} \) on \( \mathbf{c} = 4 \)
• C. \( \alpha + \beta = 60^\circ \), Projection of \( \mathbf{a} \) on \( \mathbf{c} = 6 \)
• D. \( \alpha + \beta = 90^\circ \), Projection of \( \mathbf{a} \) on \( \mathbf{c} = 7 \)

Hint:

To find the angle between vectors, use the dot product formula: \[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \] and use the projection formula for projections: \[ \text{Proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]