A First Course in Linear Algebra, 2017a

260 R n 4
3
First we want to know the total time of the swim across the river. The velocity in the direction across
the river is 3 kilometers per hour, and the river is 1 2
kilometer wide. It follows the trip takes 1/6 hour or
10 minutes.
Now, we can compute how far downstream he will end up. Since the river runs at a rate of 4 kilometers
per hour, and the trip takes 1/6 hour, the distance traveled downstream is given by 4 ( )
1
6
=
2
3
kilometers.
The distance traveled by the swimmer is given by the hypotenuse of a right triangle. The two arms of
the triangle are given by the distance across the river, 1 2 km, and the distance traveled downstream, 2 3 km.
Then, using the Pythagorean Theorem, we can calculate the total distance d traveled.
√ (2
3
d =
) 2 ( ) 1 2
+ = 5 2 6 km
Therefore, the swimmer travels a total distance of 5 6 kilometers.
♠
4.12.2 Work
The mathematical concept of work is an application of vectors in R n . The physical concept of work differs
from the notion of work employed in ordinary conversation. For example, suppose you were to slide a
150 pound weight off a table which is three feet high and shuffle along the floor for 50 yards, keeping the
height always three feet and then deposit this weight on another three foot high table. The physical concept
of work would indicate that the force exerted by your arms did no work during this project. The reason
for this definition is that even though your arms exerted considerable force on the weight, the direction of
motion was at right angles to the force they exerted. The only part of a force which does work in the sense
of physics is the component of the force in the direction of motion.
Work is defined to be the magnitude of the component of this force times the distance over which it
acts, when the component of force points in the direction of motion. In the case where the force points
in exactly the opposite direction of motion work is given by (−1) times the magnitude of this component
times the distance. Thus the work done by a force on an object as the object moves from one point to
another is a measure of the extent to which the force contributes to the motion. This is illustrated in the
following picture in the case where the given force contributes to the motion.
⃗F ⊥
θ
⃗F
⃗F ||
Q
P
Recall that for any vector ⃗u in R n ,wecanwrite⃗u as a sum of two vectors, as in
⃗u =⃗u || +⃗u ⊥