a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c<v_x, v_y> = <cv_x, cv_y>.

Students should know that multiplying a vector by a scalar is just multiplying a vector by a number. Our end result, though, is still a vector. If we multiply the components of a vector by a scalar, each of the components is multiplied by the scalar. It's sort of like our old friend the distributive property again. We've missed him.

So, what happens if the vector <10, 40> is multiplied by 0.5? Probably what you would have guessed. Each of the components is cut in half, with the result being <5, 20>.

Logically, if we take a force—say a gust of wind or a river current—and multiply it by, say 3, then whatever is effected by that gust or current will go 3 times as far. That's pretty much what all this boils down to.

So what happens if we multiply a vector v = <5, 5> by -1? The result is <-5, -5>. But hold up. If v = <5, 5> and our new vector is <-5, -5>, isn't our new vector just -v? 

Yes. That's exactly what it is. If we want to change the direction of a vector, all we need to do is multiply it by a negative scalar. Scalars can change both magnitudes and directions. Knock two birds with one stone, or two components with one scalar. (It's not a saying yet, but we're pretty sure it'll catch on.)