Question

If \(P(3,2,6)\), \(Q(1,4,5)\) and \(R(3,5,3)\) are the vertices of \(\triangle PQR\), then the measure of \(\angle PQR\) is

• A. \(90^\circ\)
• B. \(50^\circ\)
• C. \(70^\circ\)
• D. \(30^\circ\)

Hint:

If the dot product of two vectors is zero, the angle between them is \(90^\circ\).

Answer 1

The Correct Option is A

Step 1: Find vectors \(\vec{QP}\) and \(\vec{QR}\).
\[ \vec{QP} = P - Q = (3-1,\;2-4,\;6-5) = (2,-2,1) \] \[ \vec{QR} = R - Q = (3-1,\;5-4,\;3-5) = (2,1,-2) \]
Step 2: Compute the dot product.
\[ \vec{QP}\cdot\vec{QR} = (2)(2)+(-2)(1)+(1)(-2) = 4-2-2 = 0 \]
Step 3: Conclude the angle.
Since the dot product is zero, the vectors are perpendicular. Hence, \[ \angle PQR = 90^\circ \]