Understanding the Vector Identity and Its Matrix Representation

[makhoma]

vector identity??

Homework Statement

The text that I'm reading has a line that reads

$$\left(\mathbf{b}\mathbf{k}\cdot-\mathbf{b}\cdot\mathbf{k}\right)\mathbf{v}=\omega\mathbf{B}$$

and I'm not sure what it means by $\mathbf{b}\mathbf{k}$; it's clearly not the dot product nor the cross product. A line or two below it gives a matrix of the equation:

$$\left(\begin{array}{ccc}-k_{||}b & 0 & 0 \\ 0 & -k_{||}b &0 \\ k_\perp b & 0 & 0 \end{array}\right)\left(\begin{array}{c} v_x \\ v_y \\ v_z\end{array}\right)=\omega\left(\begin{array}{c}B_x \\ B_y \\ B_z\end{array}\right)$$

for $\mathbf{b}=(0,0,b)$ and $\mathbf{k}=(k_\perp,0,k_{||})$ which looks like maybe $\mathbf{b}\mathbf{k}\cdot\mathbf{v}=0$ for x and y directions but not for z??Any suggestions?

Homework Equations

unknown

The Attempt at a Solution

see above

 

[arkajad]

$$\mathbf{b}^T \mathbf{k}-(\mathbf{b}\cdot \mathbf{k})I$$

 

[makhoma]

Interesting thought there...I'll take a look at that.

 

[makhoma]

Wow that was silly easy. Thanks a bunch for your help arkajad.