$\vec{v}=2i-4j,\qquad \vec{b}=i+j$
I have no idea where to start on this.
$\vec{v}=2i-4j,\qquad \vec{b}=i+j$
I have no idea where to start on this.
First step is to draw the vectors so you can visualize possible solutions to the problem. The second step is to find a vector that is orthogonal to $\vec{b}$, such as $\vec{u} = i - j$. The thirst step is to write $\vec{v}$ in terms of $\vec{b}$ and $\vec{u}$, i.e.
$\vec{v} = \alpha \,\vec{b} + \beta \,\vec{u}$
and find scalars $\alpha$ and $\beta$. You will have to solve two equations in two unknowns.
There's a more direct route to this when you only need to express one vector relative to $b$ in this way. Take $\frac{(v.b)}{(b.b)}b$, noticing that's parallel to $b$, and add $v-\frac{(v.b)}{(b.b)}b$. You should be able to check directly that the second vector I gave is perpendicular to $b$; and obviously they add up to $v$.
The first vector is called the projection of $v$ onto $b$, and it's quite an important notion to have in hand as you're learning linear algebra.