Final answer:
To determine the angle PQR on a coordinate plane, one must calculate the vectors PQ and QR from the coordinates, find their dot product, and then use this information along with the magnitudes of the vectors to compute the cosine of the angle between them. The exact angle cannot be determined without a calculator.
Step-by-step explanation:
To find the angle ∠PQR formed by the given points P(2,5), Q(5,1), and R(0,−3) in the coordinate plane, we can utilize the dot product of vectors PQ and QR. First, we find the coordinates of vectors PQ and QR by subtracting the coordinates of the initial points from the terminal points:
- PQ = Q − P = (5 - 2, 1 - 5) = (3, −4)
- QR = R − Q = (0 - 5, −3 - 1) = (−5, −4)
The dot product of PQ and QR is given by:
PQ · QR = (3)( −5) + (−4)( −4) = −15 + 16 = 1
The magnitudes of PQ and QR are calculated as follows:
|PQ| = √(32 + (−4)2) = √(9 + 16) = √25 = 5
|QR| = √((−5)2 + (−4)2) = √(25 + 16) = √41
Now, we use the dot product formula to find the cosine of the angle θ between PQ and QR:
cosθ = PQ · QR / (|PQ| × |QR|) = 1 / (5 × √41)
Finally, we find the angle θ by taking the inverse cosine (cos−1) of this value. However, without a calculator, we cannot determine the exact angle, indicating the answer could be any angle not listed as an option (A, B, C, or D).