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: \[ \text{Proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} \]

Answer 1

The Correct Option is B

The vectors provided are: \[ \mathbf{a} = i + 3j - k, \quad \mathbf{b} = 3i - j + 2k, \quad \mathbf{c} = i + 2j - 2k \]
Step 1: Compute \( \mathbf{a} \cdot \mathbf{c} \) The dot product \( \mathbf{a} \cdot \mathbf{c} \) is computed as: \[ \mathbf{a} \cdot \mathbf{c} = (1)(1) + (3)(2) + (-1)(-2) = 1 + 6 + 2 = 9 \]
Step 2: Determine the magnitude of \( \mathbf{c} \) The magnitude of vector \( \mathbf{c} \) is calculated as: \[ |\mathbf{c}| = \sqrt{1^2 + 2^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]
Step 3: Derive the angle using the given equation The provided equation is: \[ \frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{c}|} = \frac{10}{3} \] Substituting the computed values: \[ \frac{9}{3} = 3 \] The angle \( \theta \) between \( \mathbf{a} \) and \( \mathbf{c} \) satisfies: \[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{a}| |\mathbf{c}|} \] Given \( \mathbf{a} \cdot \mathbf{c} = 9 \) and \( |\mathbf{c}| = 3 \), we first find the magnitude of \( \mathbf{a} \): \[ |\mathbf{a}| = \sqrt{1^2 + 3^2 + (-1)^2} = \sqrt{1 + 9 + 1} = \sqrt{11} \] Substituting into the cosine equation: \[ \cos \theta = \frac{9}{3 \times \sqrt{11}} = \frac{3}{\sqrt{11}} \] Therefore, \( \theta = \cos^{-1} \left( \frac{3}{\sqrt{11}} \right) \). This angle is approximately \( 45^\circ \).
Step 4: Calculate the projection of \( \mathbf{a} \) onto \( \mathbf{c} \) The projection of vector \( \mathbf{a} \) onto \( \mathbf{c} \) is defined as: \[ \text{Proj}_{\mathbf{c}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{c}}{|\mathbf{c}|} \] Using the previously calculated values: \[ \text{Proj}_{\mathbf{c}} \mathbf{a} = \frac{9}{3} = 3 \] The projection of \( \mathbf{a} \) onto \( \mathbf{c} \) is \( 3 \).