n/Example 62.o/Example 63.A shortcut example 62
.A split second after nine o’clock, the hour hand on a clock dial
has moved clockwise past the nine-o’clock position by some im-
perceptibly small angleφ. Let positivexbe to the right and posi-
tiveyup. If the hand, with length`, is represented by a∆rvector
going from the dial’s center to the tip of the hand, find this vector’s
∆x.
.The following shortcut is the easiest way to work out examples
like these, in which a vector’s direction is known relative to one
of the axes. We can tell that∆rwill have a large, negativex
component and a small, positivey. Since∆x < 0, there are
really only two logical possibilities: either∆x=−`cosφ, or∆x=
−`sinφ. Becauseφis small, cosφis large and sinφis small.
We conclude that∆x=−`cosφ.
A typical application of this technique to force vectors is given in
example 71 on p. 209.Addition of vectors given their components
The easiest type of vector addition is when you are in possession
of the components, and want to find the components of their sum.
San Diego to Las Vegas example 63
.Given the∆xand∆yvalues from the previous examples, find
the∆xand∆yfrom San Diego to Las Vegas.
.∆xt ot al=∆x 1 +∆x 2
=−120 km + 290 km
= 170 km
∆yt ot al=∆y 1 +∆y 2
= 150 km + 230 km
= 380Addition of vectors given their magnitudes and directions
In this case, you must first translate the magnitudes and direc-
tions into components, and the add the components.
Graphical addition of vectors
Often the easiest way to add vectors is by making a scale drawing
on a piece of paper. This is known as graphical addition, as opposed
to the analytic techniques discussed previously.
From San Diego to Las Vegas, graphically example 64
.Given the magnitudes and angles of the∆rvectors from SanSection 3.4 Motion in three dimensions 203