## Dot Product, Lengths, and Distances of Complex Vectors

## Problem 689

For this problem, use the complex vectors

\[ \mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 – i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 – i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 – 3i \\ 2i \end{bmatrix} . \]

Suppose $\mathbf{w}_4$ is another complex vector which is orthogonal to both $\mathbf{w}_2$ and $\mathbf{w}_3$, and satisfies $\mathbf{w}_1 \cdot \mathbf{w}_4 = 2i$ and $\| \mathbf{w}_4 \| = 3$.

Calculate the following expressions:

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

**(b)** $ \mathbf{w}_1 \cdot \mathbf{w}_3 $.

**(c)** $((2+i)\mathbf{w}_1 – (1+i)\mathbf{w}_2 ) \cdot \mathbf{w}_4$.

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

**(e)** $\| 3 \mathbf{w}_4 \|$.

**(f)** What is the distance between $\mathbf{w}_2$ and $\mathbf{w}_3$?