Determine the length of the projection of PQ onto the x-axis, with position vectors of P <3,4> and Q <7,8>

Question

Determine the length of the projection of PQ onto the x-axis, with position vectors of P <3,4> and Q <7,8>

My working out so far:

$PQ = PO + OQ$

$= -(3i + 4j) + (7i + 8j)$

$ = 4i + 4j$

Since $PQ$ is being projected onto the $x$-axis, do you choose a random vector lying on the $x$-axis, like a unit vector $(1i + 0j)$ to calculate the scalar projection with the formula?

Or the answer key provided suggests that the scalar is just the x-component of $PQ$ - so in this case it would just be $4$ since that is the $i$ component, but I don't understand why. Could someone please explain this to me?

Thanks in advance!

Answer 1

Both methods work. Using the projection formula we get the projected vector length as $$(4,4)\cdot(1,0)=4$$ Note that $(1,0)$ is already a unit vector.

Since we're projecting on the $x$-axis, it suffices to throw away the $y$-component, so we still get $4$. Indeed, using the projection formula shows that we can make this simplification.