There are bound to be several questions on SAT II Physics
that involve vector addition, particularly in mechanics. The test
doesn’t demand a very sophisticated understanding of vector addition,
but it’s important that you grasp the principle. That is, you won’t
be asked to make complicated calculations, but you will be expected
to know what happens when you add two vectors together.
The easiest way to learn how vector addition works is
to look at it graphically. There are two equivalent ways to add
vectors graphically: the tip-to-tail method and the parallelogram method.
Both will get you to the same result, but one or the other is more
convenient depending on the circumstances.
We can add any two vectors, A and B,
by placing the tail of B so
that it meets the tip of A. The
sum, A + B,
is the vector from the tail of A to
the tip of B.
Note that you’ll get the same vector if you place the
tip of B against the
tail of A. In other words, A + B and B + A are
To add A and B using
the parallelogram method, place the tail of B so
that it meets the tail of A.
Take these two vectors to be the first two adjacent sides of a parallelogram,
and draw in the remaining two sides. The vector sum, A + B,
extends from the tails of A and B across the
diagonal to the opposite corner of the parallelogram. If the vectors
are perpendicular and unequal in magnitude, the parallelogram will
be a rectangle. If the vectors are perpendicular and equal in magnitude,
the parallelogram will be a square.
Adding Vector Magnitudes
Of course, knowing what the sum of two vectors looks like
is often not enough. Sometimes you’ll need to know the magnitude
of the resultant vector. This, of course, depends not only on the
magnitude of the two vectors you’re adding, but also on the angle
between the two vectors.
Adding Perpendicular Vectors
Suppose vector A
a magnitude of 8, and vector B
perpendicular to A
a magnitude of 6. What is the magnitude of A
Since vectors A
perpendicular, the triangle formed by A
a right triangle. We can use the Pythagorean Theorem to calculate
the magnitude of A
Adding Parallel Vectors
If the vectors you want to add are in the same direction,
they can be added using simple arithmetic. For example, if you get
in your car and drive eight miles east, stop for a
break, and then drive six miles east, you will be 8
+ 6 = 14 miles east of your origin. If you drive eight miles
east and then six miles west, you will end up 8 – 6 = 2 miles
east of your origin.
Adding Vectors at Other Angles
When A and B are
neither perpendicular nor parallel, it is more difficult to calculate
the magnitude of A + B because
we can no longer use the Pythagorean Theorem. It is possible to
calculate this sum using trigonometry, but SAT II Physics will never
ask you to do this. For the most part, SAT II Physics will want
you to show graphically what the sum will look like, following the
tip-to-tail or parallelogram methods. On the rare occasions that you
need to calculate the sum of vectors that are not perpendicular,
you will be able to use the component method of vector addition,
explained later in this chapter.
||Vector A has
a magnitude of 9 and points due north, vector B has
a magnitude of 3 and points due north, and vector C has
a magnitude of 5 and points due west. What is the magnitude of the
resultant vector, A + B + C?
First, add the two parallel vectors, A and B.
Because they are parallel, this is a simple matter of straightforward
addition: 9 + 3 = 12. So the vector A + B has
a magnitude of 12 and points due north. Next, add A + B to C.
These two vectors are perpendicular, so apply the Pythagorean Theorem:
The sum of the three vectors has a magnitude of 13. Though
a little more time-consuming, adding three vectors is just as simple
as adding two.