Chapter 8.4, Problem 3CFU

To determine

To find: That the inner product of a non-zero vector and itself ever be zero.

Answer to Problem 3CFU

The explanation is given below.

Explanation of Solution

Let $a$ be any non-zero vector.

So the inner product of the vector $a$ with itself is given as:

  $\langle a,a\rangle ={\Vert a\Vert }^2$ .

As we know that the norm of a vector is zero if and only if the vector itself is a zero vector. And the vector $a$ is not a zero vector, so the norm of the vector $a$ is not zero.

Therefore the inner product of the vector $a$ with itself is not zero.

As $a$ is an arbitrary vector, so the inner product of any non-zero vector and itself cannot be zero.