Question

(a) If uand vhave the same magnitude, show that u + vand u - v are orthogonal.

(b) Use this to prove that an angle inscribed in a semicircle is a right angle (see the figure).

Short answer

Part a. The vectors $\left(\mathbf{u}+\mathbf{v}\right)and\left(\mathbf{u}-\mathbf{v}\right)$are orthogonal.

Part b. Yes, the angle inscribed in a semicircle is a right angle.

Step-by-step solution

Part (a) Step 1. Given Information

There are two vectors u and vwhich have the same magnitude. We have to show that $\left(\mathbf{u}+\mathbf{v}\right)and\left(\mathbf{u}-\mathbf{v}\right)$are orthogonal.

Part (a) Step 2. Showing the vectors are orthogonal

As we know if any two vectors are orthogonal then their dot product is equal to zero.

Let's find the dot product of$\left(\mathbf{u}+\mathbf{v}\right)and\left(\mathbf{u}-\mathbf{v}\right).$

So,

$$\begin{gathered} \left(\mathbf{u}+\mathbf{v}\right)·\left(\mathbf{u}-\mathbf{v}\right) \\ =\mathbf{u}·\mathbf{u}-\mathbf{u}·\mathbf{v}+\mathbf{v}·\mathbf{u}-\mathbf{v}·\mathbf{v} \\ ={\left|\left|\mathbf{u}\right|\right|}^2-{\left|\left|\mathbf{v}\right|\right|}^2\left[\mathbf{a}\mathbf{·}\mathbf{a}={\left|\left|\mathbf{a}\right|\right|}^2\right] \\ =0\left[\mathbf{u}\mathbf{=}\mathbf{v}\right] \end{gathered}$$

Therefore, given vectors are orthogonal.

Part (b) Step 1. Given Information

The given figure is

We have to use the figure to prove that an angle inscribed in a semicircle is a right angle.

Part (b) Step 2. Proving

To prove that an angle inscribed in a semicircle is a right angle we will use the formula of the angle between vectors.

So, $\cos \theta =\frac{\mathbf{u}\mathbf{·}\mathbf{v}}{\left|\left|\mathbf{u}\right|\right|\left|\left|\mathbf{v}\right|\right|}$

As we know that the dot product is zero

$$\begin{gathered} \cos \theta =\frac{0}{\left|\left|\mathbf{u}\right|\right|\left|\left|\mathbf{v}\right|\right|} \\ \cos \theta =0 \\ \theta ={90}^{\circ } \end{gathered}$$

Hence proved that an angle inscribed in a semicircle is a right angle.