Question

Let \( A(0,3,-3) \), \( B(1,1,1) \) and \( C(2,0,3) \) be three points in space. Then the projection of \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to

• A. \( \frac{26}{7} \)
• B. \( \frac{32}{7} \)
• C. \( \frac{34}{7} \)
• D. \( \frac{24}{7} \)
• E. \( \frac{20}{7} \)

Hint:

The sum of squares of direction cosines is always 1: \[ \alpha^2 + \beta^2 + \gamma^2 = 1 \]

Answer 1

The Correct Option is B

The formula for the projection of \( \overrightarrow{AB} \) onto \( \overrightarrow{AC} \) is given by: \[ {Proj}_{\overrightarrow{AC}} \overrightarrow{AB} = \frac{\overrightarrow{AB} \cdot \overrightarrow{AC}}{\left| \overrightarrow{AC} \right|} \] Using the given points and computing dot product, we obtain: \[ \frac{32}{7} \] Thus, the correct answer is (B).