diketahui dua titik P(4,3,5) , Q(-2,3,5) , dan R(4,3,-1) . tentukan besar sudut ∠PQR dan∠QPR beserta penjelasannya.

P = (4,3,5)
Q = (-2,3,5)
R = (4,3,-1)
Ditanya: a) Sudut PQR, b) Sudut QPR

a) Sudut PQR
QP = P - Q
QP = (4,3,5) - (-2,3,5)
QP = (6,0,0)

QR = R - Q
QR = (4,3,-1) - (-2,3,5)
QR = (6,0,-6)

QP • QR = |QP| × |QR| × cosA
(6 × 6) + (0 × 0) + (0 × -6) = √6² + 0² + 0² × √6² + 0² + (-6)² × cos A
36 = 6 × 6√2 × cosA
36 = 36√2 × cosA
36/36√2 = cosA
½√2 = cosA
45° = A

b) Sudut QPR
PQ = Q - P
PQ = (-2,3,5) - (4,3,5)
PQ = (-6,0,0)

PR = R - P
PR = (4,3,-1) - (4,3,5)
PR = (0,0,-6)

PQ • PR = |PQ| × |PR| × cosA
(-6 × 0) + (0 × 0) + (0 × -6) = √(-6)² + 0² + 0² × √0² + 0² + (-6)² × cosA
0 = 6 × 6 × cosA
0 = 36 × cosA
0 = cosA
90° = A

(A = alfa)