Question

If \( A = (0, 4, -3),\ B = (5, 0, 12),\ C = (7, 24, 0) \), then \( \angle BAC = \)

• A. \( 60^\circ \)
• B. \( \cos^{-1}\left( \dfrac{16}{\sqrt{13}} \right) \)
• C. \( \cos^{-1}\left( \dfrac{13}{38} \right) \)
• D. \( 90^\circ \)

Hint:

If the dot product of two vectors is zero, the vectors are perpendicular and form a right angle.

Answer 1

The Correct Option is D

Step 1: Use Vector Form
We want \( \angle BAC \). Use vectors \( \vec{AB} \) and \( \vec{AC} \).
\[ \vec{AB} = B - A = (5 - 0,\ 0 - 4,\ 12 + 3) = (5,\ -4,\ 15) \] \[ \vec{AC} = C - A = (7 - 0,\ 24 - 4,\ 0 + 3) = (7,\ 20,\ 3) \] Step 2: Use Dot Product Formula
\[ \vec{AB} \cdot \vec{AC} = 5 \cdot 7 + (-4) \cdot 20 + 15 \cdot 3 = 35 - 80 + 45 = 0 \] Step 3: Interpret Result
\[ \vec{AB} \cdot \vec{AC} = 0 \Rightarrow \text{Angle between vectors is } 90^\circ \] Therefore, \( \boxed{\angle BAC = 90^\circ} \)