Question

Find the value of \( p \) if the vectors \( p\hat{i} - 8\hat{j} + 5\hat{k} \) and \( 5\hat{i} + 2\hat{j} - 3\hat{k} \) are perpendicular to each other.

Hint:

The dot product is the quickest way to check for perpendicularity. Remember: Dot product \( \to \) Zero \( \to \) Perpendicular.

Answer 1

Step 1: Understanding the Concept:
Two vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular (orthogonal) if and only if their dot product (scalar product) is zero.
Step 2: Key Formula or Approach:
\( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 = 0 \).
Step 3: Detailed Explanation:
Let \( \vec{a} = p\hat{i} - 8\hat{j} + 5\hat{k} \) and \( \vec{b} = 5\hat{i} + 2\hat{j} - 3\hat{k} \).
Since the vectors are perpendicular:
\[ \vec{a} \cdot \vec{b} = 0 \] \[ (p)(5) + (-8)(2) + (5)(-3) = 0 \] \[ 5p - 16 - 15 = 0 \] \[ 5p - 31 = 0 \] \[ 5p = 31 \] \[ p = \frac{31}{5} = 6.2 \] Step 4: Final Answer:
The value of \( p \) is \( 6.2 \).