Lesson 3: Vectors II
I.
Polar to Cartesian Conversion
BVIC
Method:
1. Draw a vector sketch.
2. From the vector sketch, draw the associated triangle.
3. Find the lengths of the sides of the triangle.
4. Add signs to the values found in step #3 to get the vector components. (Signs depend on  vector
direction.
5. Check your work by drawing the result on the graph in #1. Do you get the same result?
Example 1: Consider the vector shown below
Using trigonometry and the enclosed triangles, we have
1)
2)
After determining the signs, we have the vector components
Example 2: Convert

A  5m  36.9  into Cartesian form.
Example 3: Write the vector shown below in Cartesian form.
Y
8m
60
X
II.
Cartesian to Polar Representation
Method:
1. Draw a vector sketch.
2. From the vector sketch, draw the associated triangle.
3. Find the magnitude using the Pythagorean theorem
4. Find the angle of the triangle using trigonometry (tangent function) and the triangle in step #2.
5. You can get the final angle with respect to the +x-axis from the sketch in #1 using geometry. If
you don't draw an angle on a sketch then I will assume that it is measured counter-clockwise from
the +x-axis.

Example: Write A   2 î  5 ĵ in Cartesian form.
1)
2)
3)
4)
5)
III. Zero Vector
A vector is zero if and only if its magnitude (length) is _______________________
or equivalently
ALL of its _______________________ are _______________________.
IV. Multiplication of Vectors
Multiplication for vectors is more complicated than for scalars. In fact, there is three different types of
vector multiplication:
a) Multiplication of a Vector by a Scalar (previously covered)
b) Scalar Vector Multiplication (Dot Product)
c) Vector (Cross) Product
V.
Scalar Vector Multiplication
1.
Notation
2.
Result is a scalar whose value is found by the formula
3.
Graphical Interpretation
From the diagrams, we see that the dot product of two vectors, A and B, can be viewed either as the
length of vector A times the component of vector B along vector A's direction.
or equivalently
the length of vector B times the component of vector A along vector B.
Example 1: Find the dot product of the two vectors shown below:
4.
Cartesian Form of the Dot Product
Our existing form for the dot product is very inconvenient for dealing with vectors that already in
Cartesian form. We can obtain a useful form for dealing with vectors in Cartesian form simply
multiplying the unit vectors like ordinary variables in Algebra and then using our existing dot
product formula to evaluate the results.
Given
Proof:


A  A x î  A y ĵ  A z k̂ and B  B x î  B y ĵ  B z k̂ , the dot product of the two vectors is


Example: Given A  3 î  2 ĵ and B   4 î  3 ĵ , find the dot product.
5.
Application of the Dot Product
The dot product has many important applications. In fact, you have already seen one result of the dot
product (Pythagorean Theorem)!!
a)
Calculating A Vector's Magnitude
b)
Creating A Unit Vector
If you need to create a unit vector that points in the same direction as another vector
do the following:

A , then you can
c)
Finding the Component of a vector
Physicists and engineers often need to express vectors in coordinate systems other than just
Cartesian. We can determine the components of the vector along any set of unit vectors using the
dot product.
Example: Directional Cosines
Using the Cartesian unit vectors, we can develop the directional cosines method of expressing a three
dimensional vector. This for is very useful for solving engineering problems (used in Engineering
Principles I).
VI
Vector Cross Product
1.
Notation
2.
The result is a ______________________.
i) The magnitude is given by the formula
The magnitude of the cross product of two vectors A and B can be viewed as the length of vector A
times the component of vector B that is perpendicular to vector A
or equivalently
the length of vector B times the component of vector A that is perpendicular to vector B.
ii) The direction is given by the ______________ _______________ _________________ and is
________________________ to both ______________ and _______________.
3.
Unlike the dot product, the vector cross product does NOT commute.
Example 1: Find
  
C  A  B for the vectors drawn below:
Y
X
Solution:


Example 2: Find C for the vectors below if B  4 î  3 ˆj

B
5

A
4.
Working With 3-D Vectors
The cross product can be calculated easily for three-dimensional vectors using any one of three
equivalent processes:
i) Do the algebra with the unit vectors and then use our previous results to evaluate the results
ii) Use the determinant to evaluate the cross product
iii) Use cyclic perturbation to solve for the cross product.


Example: Given A  3 î  2 ĵ  1 k̂ and B  - 2 î  0 ĵ  4 k̂ , find
  
C  AB
5.
Other Uses of the Cross Product
The cross product can be used to find the directional area bound by two vectors:
By combining the dot product and cross product, we can represent any vector in 2-dimensions as