## Inner Products, Lengths, and Distances of 3-Dimensional Real Vectors

## Problem 687

For this problem, use the real vectors

\[ \mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} . \]
Suppose that $\mathbf{v}_4$ is another vector which is orthogonal to $\mathbf{v}_1$ and $\mathbf{v}_3$, and satisfying

\[ \mathbf{v}_2 \cdot \mathbf{v}_4 = -3 . \]

Calculate the following expressions:

**(a)** $\mathbf{v}_1 \cdot \mathbf{v}_2 $.

**(b)** $\mathbf{v}_3 \cdot \mathbf{v}_4$.

**(c)** $( 2 \mathbf{v}_1 + 3 \mathbf{v}_2 – \mathbf{v}_3 ) \cdot \mathbf{v}_4 $.

**(d)** $\| \mathbf{v}_1 \| , \, \| \mathbf{v}_2 \| , \, \| \mathbf{v}_3 \| $.

**(e)** What is the distance between $\mathbf{v}_1$ and $\mathbf{v}_2$?