Question

Prove that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\). (This is Theorem 10.19.)

Short answer

It is proven that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\).

Step-by-step solution

Step 1. Given Information

We have to prove that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\).

Orthogonal vectors are those vectors that are perpendicular to each other.

Step 2. Prove

To prove that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\). We will use the formula for the angle between two vectors \(u\) and \(v\) which is \(u\cdot v=\left\|u \right\|\left\|v \right\|cos\theta\).

Since \(u\neq 0,v\neq 0\). So, \(u\cdot v=0 \) is possible only if \(cos\theta=0\).

Thus, \(u · v = 0\) if the vectors are orthogonal.