Mathematical Investigations IV
Name:
Vectors - Getting To the Point
Projections
Recall the following:
 the Geometrical Interpretation of the dot product states that v  w = |v| |w| cos
 |w| 2 = w  w
w
ˆ 
 w
.
|w|
Let’s see how to find the projection
of one vector onto another.

(2)
(3)
v

Consider the diagram at the right.
w
We want to find the projection of v onto w. Drop a perpendicular segment from the tip of v
onto the line containing w. Take a look at the right triangle formed: | v | is the length of the
hypotenuse, and the perpendicular (dashed) segment is the side opposite the angle . The
adjacent side, along w, is called the vector projection of v onto w, written projw v.
v
v


projwv
w
w
Use right triangle trigonometry to find the length of projw v in terms of | v | and  .
| projw v | =
Now, we have the length of projw v but we need to know its direction. It is in the same direction
as what other vector? How do we denote the unit vector in this direction?
Note that if we multiply a unit vector by a specific length, we will have a vector with this length
in the direction of the unit vector. Use this to find the vector projw v in terms of | v |, , and ŵ .
projw v =
Vectors 6.1
Rev F09
Mathematical Investigations IV
Name:
Now, from the Geometrical Interpretation of the Dot Product, solve for cos .
cos =
Substitute this, along with other information from (2) and (3) on the top of page 6.1, into the
expression for projwv and simplify.
projwv 
Projection formula
v  w 
projw v  
w
w  w
Using our formula for projections and our formula for dot products, we can compute projections
just knowing the coordinates with a few simple multiplications, additions, and divisions.
1.
Let p = <3, 1> and q = <2, 6>. Find projq p and projp q and sketch them on the graph
below.
Vectors 6.2
Rev F09
Mathematical Investigations IV
Name:
2.
Sketch the vectors v = 3 iˆ + 4 ĵ and w = –2 iˆ – ĵ .
First find the angle between v and w.
Next, find the projections projv w and projw v using the formula.
a)
projv w
b)
projw v
Now sketch projv w and projw v. What do you notice? Formulate a conjecture about projv w
when the angle between v and w is obtuse.
Vectors 6.3
Rev F09
Mathematical Investigations IV
Name:
3.
When does projw v  0, 0 ?
4.
Under what conditions does projw v = w?
5.
What is the vector projection of iˆ onto iˆ ?
6.
What is the vector projection of v onto –v?
7.
Explain why the formula for determining the projection of one vector onto another gives
the correct result for any angle  (0    180) between the vectors?
Vectors 6.4
Rev F09