Calculus and Vectors – How to get an A+
7.5 Scalar and Vector Projections
A Scalar Projection
r
r
The scalar projection of the vector a onto the vector b is a
scalar defined as:
r
r
r
r r
SProj (a onto b ) =|| a || cosθ where θ = ∠(a , b )
Ex 1. Given two vectors with the magnitudes
r
r
|| a ||= 10 and || b ||= 16 respectively, and the angle
between them equal to θ = 120° , find the scalar
projection
r
r
a) of the vector a onto the vector b
r
r
b) of the vector b onto the vector a
B Special Cases
r
r
Consider two vectors a and b .
r
r
r
r
r
a) If a ↑↑ b ( cosθ = 1 ), then SProj (a onto b ) =|| a ||
r
r
r
r
r
b) If a ↑↓ b ( cosθ = −1 ), then SProj (a onto b ) = − || a ||
r
r
r r
c) If a ⊥ b then SProj (a onto b ) = 0
r
Ex 2. Find the scalar projection of the vector a onto:
C Dot Product and Scalar Projection
r
r
Recall that the dot product of the vectors a and b is defined
as:
r r r r
a ⋅ b =|| a || || b || cosθ
r
r
So, the scalar projection of the vector a onto the vector b
can be written as:
r r
r
r
r
a ⋅b
SProj (a onto b ) =|| a || cosθ = r
|| b ||
Note:
For a Cartesian (Rectangular) coordinate system, the scalar
r
components a x , a y , and a z of a vector a = (a x , a y , a z ) are
r
r
the scalar projections of the vector a onto the unit vectors i ,
r
r
j , and k .
Proof:
r
r ar ⋅ i (a x , a y , a z ) ⋅ (1,0,0)
r
SProj (a onto i ) = r =
= ax
1
|| i ||
r
Ex 3. Given the vector a = (2,−3,4) , find the scalar
projection:
r
r
a) of a onto the unit vector i
D Vector Projection
r
r
The vector projection of the vector a onto the vector b is a
vector defined as:
r
r
r
r
b
VProj (a onto b ) =|| a || cosθ r
|| b ||
E Dot Product and Vector Projection
r
The vector projection of the vector a onto the vector
r
b can be written using the dot product as:
r r
r (ar ⋅ b )b
r
VProj (a onto b ) = r 2
|| b ||
Note:
For a Cartesian (Rectangular) coordinate system, the
r
r r
r
r
r
vector components a x = a x i , a y = a y j , and a z = a z k of
r
a vector a = (a x , a y , a z ) are the vector projections of the
r
r r
r
vector a onto the unit vectors i , j , and k .
a) itself
r
b) the opposite vector −a
r r
r
b) of a onto the vector i − j
r
r
r r
r
c) of a onto the vector b = −i + 2 j + k
r
r
d) of the unit vector i onto the vector a
7.5 Scalar and Vector Projections
©2010 Iulia & Teodoru Gugoiu - Page 1 of 2
Calculus and Vectors – How to get an A+
r
r
Ex 4. Given two vectors a = (0,1,−2) and b = (−1,0,3) , find:
r
r
a) the vector projection of the vector a onto the vector b
Ex 5. Find an expression using the dot product of the
r
vector components of the vector a along to the vector
r
b and along to a direction perpendicular to the direction
r
of the vector b but in the same plan containing the
r
r
vectors a and b .
r
r
b) the vector projection of the vector b onto the vector a
r
r
c) the vector projection of the vector a onto the unit vector k
r
r
d) the vector projection of the vector i onto the vector a
Reading: Nelson Textbook, Pages 390-398
Homework: Nelson Textbook: Page 399 # 6, 11, 13, 14
7.5 Scalar and Vector Projections
©2010 Iulia & Teodoru Gugoiu - Page 2 of 2