Chapter 3 Summary
AP Physics C
Vectors and Scalars
A scalar is a quantity that has only magnitude – i.e. speed & Distance are scalars
A vector is a quantity that has both magnitude and direction – i.e. velocity, and displacement.
For example a car has a speed of 50 miles/hour which is a scalar quantity, but the car’s velocity is 50
miles/hour due north, or 50 miles per hour 30 degrees north of west, these are vector quantities.
In physics a negative sign (-) denotes direction, not magnitude. Hence if we declare North to be the
positive direction on the vertical axis, then south is the negative direction. If east is positive
horizontally, then west is negative.
Addition of Vectors
Example if vector a is 50 meters to the east and vector b is 30 meters to the north then vector b + vector
a is determined as shown in figure A, whereas vector b – vector a is shown in figure B:
b
c
a
a
Figure A
b
c
Figure B
So the resultant vector, labeled C is determined using the quadratic equation c2= a2 + b2 and the direction
is given using tan  = b/a or a/b depending on whether you choose to find the angle using the upper or
lower triangle, which are equivalent to each other. Hence vector c is 58.3 meters @ 31 N of E.
Please note that the negative of vector a simply points in the opposite direction and hence the resultant
simply points in the opposite direction as well, giving a resultant of 58.3 meters @ 31 N of W.
Adding Multiple Vectors
Using the same right triangle geometry, you can take a single vector, which points at an angle to any of
the 4 major directions and break it into its component vectors. Hence if I have a vector C, which is 40
meters long and points 30 degrees north of West, I can break it down using the trigonometric functions
sin and cos into an equivalent pair of components which point in the north and west direction.
N = 40 Sin 30 = 20 and W = 40 Cos 30= 34.4
40 m
Hence several vectors can be added together by:
1. breaking them into components
2. Adding or subtracting the 4 directions to produce a
two vector system
3. Using Pythagorean theorem to add the vectors to
produce a single resultant.
4. Using the tangent function to find the resultant angle.
30
W
Figure C
N
Vector Multiplication
There are three kinds of vector multiplication:
1. Multiplication of a vector by a scalar – this produces a vector. For example if I have a vector
which is 10 meters long which points 30 north of west and I multiply it by the scalar 5, I get a
vector 50 meters long which points in the same direction.
2. Multiplication of two vectors to produce a scalar. – This is the dot product of two vectors.
By definition ab = ab cos  where a is the magnitude of vector a and b is the magnitude of
vector b and  is the angle between them. Example 1:if vector a is 5 units due north, and vector
b is 6 units due east. Then the dot product is (5)(6) cos 90. Since cos 90 is zero, the dot product
vanishes or equals zero. Example 2: If vector b were six units at 30 east of north then the dot
product would be (5)(6) cos 30 = 25.8. Often it is useful to express vectors in terms of their i, j,
and k components, also called unit vectors. Since every vector can be broken into components
as described above, all vectors can be expressed as a combination of unit vectors with the i
component lying along the x axis, the j along the y axis, and the k along the z axis. Making all
vectors three dimensional. Hence, looking at figure A, above vector a could be written as 50i
and vector b as 30j and vector c as 50i + 30j. Whereas the resultant in figure b would be written
as 50i – 30j. The vectors in figure C would be –34.4i + 20j. In all three cases, the z component is
0 and hence not written.
Rule for vector multiplication:
Given that ab = ab cos  and cos 90 = zero, vectors which are perpendicular to each other
do not produce a dot product. Hence i j = 0, i k = 0, j  k = 0. Only like unit vectors
produce a product, hence i i = 1, j j = 1, k  k = 1. so if I multiply the vectors:
A = 3i+2j+3k and B = 4i-2j + 3k what I get is:
(3i)(4i) + (3i)(-2j) + (3i)(3k) +(2j)(4i) + (2j)(-2j) + (2j)(3k)+ (3k)(4i) + (3k)(-2j) + (3k)(3k)
12
+
0
+
0
+
0
+ -4
+
0
+
0
+
0
+
9
= 17
which is a scalar product. Also, I can determine the magnitude of vector a or vector b if they are
written in i ,j,k form using a three dimensional Pythagorean theorem: magnitude of a =
(ai ) 2  (aj ) 2  (ak ) 2 so the magnitude of vector a in the example above would be
(3i ) 2  (2 j ) 2  (3k ) 2  22
3. Multiplication of two vectors to get a vector – this is the cross product of two vectors. By
definition a x b = ab sin  where a and b are the magnitude of vectors a and b and  is the angle
between them.
Rule for cross products: Since sin 0 = 0, vectors which lie on the same axis have a cross
product of zero. Hence i x i =0, j x j = 0 and k x k =0. Also i x j = k, j x k = i, k x i = j.
Conversely, j x i = -k, k x j = -i, and i x k = -j.
So using vectors a and b from the example above a x b =:
(3i)(4i) + (3i)(-2j) + (3i)(3k) +(2j)(4i) + (2j)(-2j) + (2j)(3k)+ (3k)(4i) + (3k)(-2j) + (3k)(3k)
0 +
-6k
12i + 3j - 14k
+ -9j
+ -8k
+ 0
+
6i
+
12j
+ 6i
+
0
=