Prove some relations between three given vectors in Rn

Let $A,B,C \in \mathbb{R}^n$ all be nonzero. Further, let the angle between $A$ and $C$ be equal to the angle between $B$ and $C$ . Prove that $C$ is orthogonal to

$\lVert B \rVert A - \lVert A \rVert B.$

Proof. Let $\theta$ be the angle between $A$ and $C$ which is equal to the angle between $B$ and $C$ . Then we have

$\cos \theta &= \frac{A \cdot C}{\lVert A \rVert \lVert C \rVert} = \frac{B \cdot C}{\lVert B \rVert \lVert C \rVert}.$

This implies,

$A \cdot C = \lVert A \rVert \lVert C \rVert \cos \theta \quad \text{and} \quad B \cdot C = \lVert B \rVert \lVert C \rVert \cos \theta.$

Then, we have

$\begin{align*} C \cdot \left( \lVert B \rVert A - \lVert A \rVert B \right) &= (A \cdot C) \lVert B \rVert - (B \cdot C) \lVert A \rVert \\[9pt] &= \lVert A \rVert \lVert B \rVert \lVert C \rVert \cos \theta - \lVert A \rVert \lVert B \rVert \lVert C \rVert \cos \theta \\[9pt] &= 0. \end{align*}$

Hence, $C$ and $\lVert B \rVert A - \lVert A \rVert B$ are orthogonal $. \qquad \blacksquare$

Stumbling Robot

Prove some relations between three given vectors in Rⁿ

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