**Scalar and vector projections**

Given two vectors, can we ask how much of one vector is pointing in the direction of the other? We can certainly ask how much of the vector is pointing in the direction – the answer is just 5. You can think of this as projecting the vector onto the -axis and asking for its projected length. Similarly we can ask about the projection of a vector into any arbitrary direction. This is illustrated in figure \ref{vec6}. Imagine having a light perpendicular to shining towards it. There is a shadow of the vector cast on the line of . This is the scalar projection of in the direction of , also called the component of in the direction of . When you are looking at this, clearly the size of is unimportant, so you can think of an infinite line stretching in both directions parallel to .

To calculate this quantity, which we call it is clearly just:

Check to make sure that you understand why the last equality is so. We can see that this therefore has nothing to do with the length of as we stated before, just its direction.

Here we have just asked how much of is pointing in the direction of . We can also define a vector of this length, in the direction of . This is simply the quantity we already have multiplied by a unit vector in the direction of and thus given by:

**Hint: Make sure that you can derive these quantities (the comp and proj) yourself. Don’t just remember the formulae but understand where they come from!**

Mathemafrica - UCT MAM1000 lecture notes subject listOctober 3, 2015 at 11:05 am[…] MAM1000 lecture notes part 41 – Scalar and vector projections […]