Name ___________________
Worksheet 9.5
Vector Projections Overview
Due Wed Jan 28
Unit Vectors
1 
A unit vector is a vector with magnitude 1. For example, the vector iˆ    is a very famous
0 
unit vector, the unit vector in the direction of the x axes. And it should be clear that iˆ  1
1
(a) What is the unit vector with direction of 45°? Note that it is not   since
1
1
1  2

5
(b) Give a unit vector in the direction of  
12 
12 
(c) Find the unit vector in the direction  4 
 6 
Unit Vector Notation
3
1
In a different representation, vectors such as a    and b    can be written as
4
 2
a  3iˆ  4 ˆj and b  1iˆ  2 ˆj , where iˆ and ĵ are unit vectors in the x and y directions.
What is the dot product of a and b ? Carry out the dot product on the vectors written in terms of
the unit vectors. In other words “F.O.I.L.” the two vectors, and try to get the correct answer.
Dot Products, Magnitudes, and Simple Projections
5
 4
1 
0
Consider the vectors A    , B    , iˆ    , and ˆj   
12
 3
0
1 
(a) Find A  B
(b) Find A  iˆ
(c) Find A  A
(d) Find the magnitude of A
(e) Give the magnitude C of an arbitrary vector C in terms of vector notation and dot products
only.
2
(f) This should be even simpler than finding C : Give C , the square of the magnitude.
(g) Is A  iˆ a scalar or a vector? How about ( A  B )iˆ ?
(h) Find the vector ( A  iˆ)iˆ and sketch this vector along side A . This vector is the projection of
A in the iˆ direction, thus in the positive x-direction as well.
(i) Come up with a simple vector formula for projection of B in the positive y-direction. Include
a sketch.
Projections in Any Direction
Now we will try to project a vector in any arbitrary direction. Consider the vector that represents
the velocity of a airplane. This airplane is going 2 km/min East and 7 km/min North. In other
words, v  iˆ  2 and v  ˆj  7 . We will try to answer this question: How fast is the plane going
in the Northeast direction?
(a) Find the vector v and sketch this velocity vector.
(b) Find a unit vector n̂ that points in the “Northeast” direction (exactly). Make sure that
nˆ  nˆ  1 . In other words, make sure your unit vector really is a unit long.
(c) Give the simple vector equation for the projection (call it p ) of v in the n̂ direction. It may
be helpful to refer to part (g) of the previous part.
(d) Find the magnitude of this projection (which is the answer to the question of how fast the
plane is going in the Northeast direction) and also find the vector itself.
(e) Sketch v , nˆ , and p together.
Projection Example
5
 4
Let us consider our original vectors A    and B    . Think of airplane Andy as traveling
12 
3
5 km/min East and 12 km/min North, while airplane Bobby is traveling 4 km/min East and 3
4
km/min North. Now let us try to answer this question: Which plane is going in the   direction
3
more rapidly? Put another way, is airplane Andy beating airplane Bobby at airplane Bobby’s
own game?
(a) Sketch A and B and try to guess the right answer just by looking.
4
(b) Find b̂ , the unit vector in the   direction. Add this to the sketch above.
3
(c) Write b̂ in terms of B only using vector notation (you can also use a square root sign).
(d) Write the simple vector equation for the vector (call it p ) projection of A in the b̂ direction.
(e) Eliminate b̂ to get a very elegant equation for the projection of A in the B direction (this
should look similar to what you have seen in the textbook)
(f) Use this equation to find p . Sketch above.
4
(g) How fast is plane Andy going in the   direction? How about Bobby? Which one is going
3
faster in this direction? By what percentage is the “faster” plane going faster, assuming speed is
4
measured only in the   direction?
3
Some Projection Practice
3
5
1
Consider the vectors A    , B    , C   
 1
 2
3
(a) Find the projection of A along the C direction. Include a sketch.
(b) Find the projection p of A along the B direction. Add this to your sketch.
(c) Find the projection of this p vector in the direction of C . Add your work to the sketch.
(d) Are your answers to part (a) and (c) the same? Should they be?
2
4


(e) Find the projection of A  3 along B   2  . Sketch if you can.
 
 1 
12 