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Projections
P. Danziger
1 Components and Projections
A A A A A A
``
θ
u A v
``
projvu
Given two vectors u and v, we can ask how far we will go in the direction of v when we travel
along u. The distance we travel in the direction of v, while traversing u is called the component of
u with respect to v and is denoted compvu. The vector parallel to v, with magnitude compvu, in
the direction of v is called the projection of u onto v and is denoted projvu.
So, compvu = ||projvu||
Note projvu is a vector and compvu is a scalar. From the picture compvu = ||u|| cos θ
We wish to find a formula for the projection of u onto v.
Consider u · v = ||u||||v|| cos θ
Thus ||u|| cos θ = u^ ·^ v ||v||
So compvu = u · v ||v||
The unit vector in the same direction as v is given by v ||v||
. So
projvu =
u · v ||v|| 2
v
Example 1
- Find the projection of u = i + 2j onto v = i + j.
u · v = 1 + 2 = 3, ||v|| 2 =
projvu =
u · v
||v||^2
v =
(i + j) =
i +
j
7.7 Projections P. Danziger
- Find projvu, where u = (1, 2 , 1) and v = (1, 1 , 2)
u · v = 1 + 2 + 2 = 5, ||v|| 2 =
12 + 1^2 + 2^2
So, projvu =
- Find the component of u = i + j in the direction of v = 3i + 4j.
u · v = 3 + 4 = 7, ||v|| =
32 + 4^2 =
compvu =
u · v
||v||
- Find the components of u = i + 3j − 2 k in the directions i, j and k.
u · i = 1, u · j = 3, u · k = − 2 ,
||i|| = ||j|| = ||k|| = 1
So compiu = 1, compju = 3, compku = − 2.
So the use of the term component is justified in this context.
Indeed, coordinate axes are arbitrarily chosen and are subject to change.
If u is a new coordinate vector given in terms of the old set then compuw gives the component of the vector w in the new coordinate system.
Example 2 If coordinates in the plane are rotated by 45o, the vector i is mapped to u = √^1 2 i + √^1 2 j, and
the vector j is mapped to v = − √^1 2 i + √^1 2 j. Find the components of w = 2i − 5 j with respect to the new coordinate vectors u and v. i.e. Express w in terms of u and v.
@
@
@
@
@
@
@
@
@
w @ w
i
j u
6
6
w · u =
, w · v =
. ||u|| = ||v|| = 1
So
compuw =
, compvw =
and
w =
u +
v
