projection vector of u=(2,3,1) along (1,4,5) is
Answers
Answer:
Projections
P. Danziger
1 Components and Projections
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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
1. Find the projection of u = i + 2j onto v = i + j.
u · v = 1 + 2 = 3, ||v||2 =
√
2
2
= 2
projvu =
u · v
||v||2
v =
3
2
(i + j) = 3
2
i +
3
2
Step-by-step explanation:
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