How do you rearrange equations with dot products in them?

Question

How can I go about rearranging an equation similar to this...

$$\left(\pmatrix{-3\\0\\1}+ t \pmatrix{1\\4\\7} \right) \cdot n - a = 0$$

The issue I'm having is manipulating dot products

Answer 1

The dot product is distributive over vector addition. This is easy to see because the dot product is essentially a sum of component-wise multiplications, and vector addition works component-wise.

That is, $$a\cdot(b+c) = a\cdot b + a\cdot c.$$

Answer 2

It helps to know that the dot product is an inner product, which means it's linear, symmetric and positive definite. So the rules to remember are:

$$a(u\cdot v)=au\cdot v=u\cdot av\\ u\cdot(v+w)=(u\cdot v)+(u\cdot w)\\u\cdot v=v\cdot u\\u\cdot u> 0\:\:\mbox{if and only if }u\neq 0$$