Question

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

Hint:

Two vectors are perpendicular if their dot product is zero.

Answer 1

Step 1: Condition for perpendicular vectors.
Two vectors are perpendicular if and only if their dot product is zero. Therefore, we need to find \( p \) such that: \[ (\vec{p} \hat{i} - 8 \hat{j} + 5 \hat{k}) \cdot (5 \hat{i} + 2 \hat{j} - 3 \hat{k}) = 0 \] Step 2: Computing the dot product.
The dot product of the two vectors is: \[ \vec{v}_1 \cdot \vec{v}_2 = p \cdot 5 + (-8) \cdot 2 + 5 \cdot (-3) \] \[ = 5p - 16 - 15 \] \[ = 5p - 31 \] Step 3: Setting the dot product equal to zero.
For the vectors to be perpendicular, we set the dot product equal to zero: \[ 5p - 31 = 0 \] Solving for \( p \): \[ 5p = 31 \quad \Rightarrow \quad p = \frac{31}{5} \] Step 4: Conclusion.
Thus, the value of \( p \) is \( \frac{31}{5} \).