Vector Projections
A vector projection represents a "shadow" of a vector onto a flat surface.
Sep 12 ­ 9:59 PM
Vector Projections ­ A familiar topic with a new twist
F
θ
D
F cosθ
Projection of F onto D
Where have we seen parts of this type of concept before?
(projection)
May 5­12:13 AM
Calculating SCALAR Projections
The scalar projection of "a" onto "b" is
The scalar projection of "b" onto "a" is
In general, For R3, May 5­12:22 AM
Review ­ Unit Vector Concept
The Unit Vector of "a" is given by:
May 5­12:28 AM
Vector Projection of "a" onto "b"
The scalar projection of "a" onto "b" is
The Unit Vector of "b" is given by:
To create the VECTOR projection of "a" onto "b" we combine the two idea together.
Vector Projection of "a" onto "b" scalar multiplier
(magnitude)
Unit Vector
(direction)
Since
May 5­12:33 AM
Scalar projection of vector a onto b is ON, where
ON = a cos
See p390 for summary.
The formulas are the same as for the vector projections ­ just use the first part ­ not the vectors.
scalar projection just scalar projection just Feb 23­12:46 PM
Find the vector projection of "u" onto "v" for the given vectors.
Given and Method One ­ Common Sense and Trig
Method Two ­ Dot Product and Common Sense
Method Three ­ Dot Product and Formula
or
May 7­10:25 AM
Find the vector projection of "u" onto "v" for the given vectors.
Given and or
May 7­10:25 AM
Direction Angles in R2
y
(x,y)
h
y
θ
x
x
where θ is measured from the vector "h" to the x axis.
Direction Angles in R3
y
P(x,y,z)
There are THREE Right Triangles that can be made from the given vector to each one of the x, y and z axis.
Vector to X axis (angle α)
O
x
z
Vector to Y axis (angle β)
Vector to Z axis (angle γ)
(component of OP along this axis)
(component of OP along this axis)
May 5­12:45 AM
Feb 23­1:00 PM
(component of OP along this axis)
Consolidation Questions
p398 # 1, 3, 6, 7ac, 11, 13
May 7­10:33 AM