Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
Example 1.3.12
Use vector methods to prove that an angle inscribed in a semicircle is necessarily a right angle.
Solution
Angle PRQ is inscribed in the semicircle shown in Figure 1.3.12(a). Vectors A and −A are along the diagonal, and vector B, connecting points O (the center of the circle) and R, is along a radius.
Vectors A and B have length r, the length of the radius of the circle whose center is O.
The vectors B+A and B−A are along the line segments PR and QR, respectively. The angle between these two vectors is ∠PRQ.
The cosine of this angle is found by the following calculation.
Figure 1.3.12(a) Angle inscribed in semicircle
cos∠PRQ
=(B+A)·(B−A)B+A B−A
=B·B−A·AB+A B−A
=B2−∥A∥2B+A B−A
=r2−r2B+A B−A
=0
Since its cosine is zero, the measure of ∠PRQ is π/2 so the angle is a right angle.
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