How can I calculate this term?

Question

How do I evaluate a term of the form

$$\frac{\langle v_1, x_2 \rangle}{\Vert v_1 \Vert}?$$

I know that the denominator is the norm. However, how can I calculate the numerator? In my example, $v_1= (1, 2, 3)$ and $x_2 = ( 1, -2, 3)$.

Thanks.

Answer 1

The expression in the numerator is the so-called standard inner product in Euclidean space of two vectors, which can be evaluated as follows:

$$\left\langle \begin{pmatrix} a \\ b \\ c \end{pmatrix}, \begin{pmatrix} x \\ y \\ z \end{pmatrix} \right\rangle = ax + by+cz.$$ Note that with this notation, we can define $\Vert v \Vert = \sqrt{\langle v, v \rangle}.$ Substituting your specific values, we obtain $$\frac{\langle v_1, x_2\rangle}{\Vert v_1 \Vert} = \frac{1 -4 + 9}{\sqrt{1 + 4 + 9}} = \frac{6}{\sqrt{14}}.$$