High School: Number and Quantity
Vector and Matrix Quantities HSN-VM.B.5b
b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Now, let's talk direction. Tim is in a canoe. If he were in standing water, Tim would be rowing at a rate of 8 mph. (Is that good? We don't know whether this makes Tim a small child or an Olympic champion.) Either way, let's say that at the end of an hour, Tim has covered 10 miles. What can we determine about the current?
Well, Tim moved a lot faster than he rowed. So we can assume that the current helped Tim—that it pushed him along towards his goal. So the current must have been going in the same direction as Tim.
This means that if v is the vector of Tim's rowing, the current travels in the same direction as v multiplied by a scalar c > 0.
But what if we change the story? Tim is still rowing at the rate of 8 mph. Only this time, at the end of an hour, Tim has covered only 5 miles. What do we know about the direction of the current? Well, it was slowing Tim down! That means the scalar for the current (relative to Tim's vector) is c < 0 because the vector goes against v.
Basically, if the product of a scalar and a vector results in a direction opposite of the original vector, then the scalar must be negative.