1. Vector Projections
Dr. Shildneck

2. Projection of a Vector onto Another Vector
We know how to add to vectors to obtain a resultant vector. We now want to reverse the process by expressing two vectors in terms of two orthogonal vectors. One reason to do this is to determine the amount of force applied in a particular direction.

3. Example of Forces
For example, imagine a boat and trailer on a ramp. The force due to gravity (Fg) pulls straight down on the boat. Part of this force pushes/pulls the boat down the ramp (F1). The other part of the force presses/holds the boat against the ramp (F2).
Note: Since vectors are
Equal when they have the
Same magnitude and direction,
We can redraw F1 and F2 as
Needed to form right triangles. F1
Since F1 and F2 are perpendicular
(orthogonal), we can break Fg down
Into these vector component forces. F2 Fg

4. Projection of a Vector onto Another Vector
Given two non-zero vectors v and w with the same initial point and an angle between them (Ѳ), a third vector, called the vector projection of v onto w, is formed in the figures. This vector is denoted as projwv.
Vector Projection with an Acute Angle
Vector Projection with an Obtuse Angle v w v Ѳ Ѳ w
projwv
projwv
First, a line is drawn through w.
Next, a line segment is dropped from the terminal point of v onto the line through w to form a right angle.
Finally, a vector is drawn from the initial point of w and v to the point where the segment intersects the line.
Note that the projection of v onto w does not change as a result of the angle.

5. Projection of a Vector onto Another Vector
Our goal, then, is to determine an expression for projwv. We begin by using it’s magnitude ||projwv||.
Using Soh-Cah-Toa…
𝑐𝑜𝑠𝜃= | projwv | | v |
multiplying
| v |𝑐𝑜𝑠𝜃=| projwv |
Substituting, we get…
Now, we also know…
v∙w=| v || w |𝑐𝑜𝑠𝜃
v∙w=| w || v |𝑐𝑜𝑠𝜃
v∙w=| w || projwv |
v∙w | w | =| projwv |
This is the magnitude of the
projection of v onto w. This is called
the Scalar Projection.
Then, dividing we get…

6. Projection of a Vector onto Another Vector
Our goal, still, to determine an expression for the vector projwv, requires us to use the magnitude to determine a vector in the same direction as w.
First, find the unit vector in the same direction as w.
Unit Vector w = w | w |
Now multiply by the magnitude we want:
| projwv |∙ w | w |
This, then is the vector in the direction of w with magnitude of the projection. Simplifying…
projwv=| projwv |∙ w | w |
= v∙w | w | w | w |
= v∙w | w | 𝟐 w
This is the VECTOR
projection of v onto w.

7. Projection of a Vector onto Another Vector
If v and w are two nonzero vectors, the vector projection of v onto w is Projwv = 𝐯∙𝐰 𝐰 2 𝐰

8. Example 1
If v = 2i + 4j and w = -2i +6j , find the vector projection of v onto w.

9. Determining the Vector Components of v
Next we want to use the vector projection of v onto w to express v as the sum of two orthogonal vectors.
Let v and w be two nonzero vectors. Vector v can be
expressed as the sum of two orthogonal vectors v1 and v2,
where v1 is parallel to w, and v2 is orthogonal to w.
𝐯1=Projwv = 𝐯∙𝐰 𝐰 2 𝐰
and
𝐯2= v – v1
Since v = v1 + v2, the vectors v1 and v2 are called the vector components of v.
The process of expressing v as v1 + v2 is called the decomposition of v.

10. Example 2
If v = 2i – 5j and w = i – j, decompose v into orthogonal vectors where one is parallel to w.