Calculate projections of vectors

In summary, the conversation discusses computing projectors P1 and P2 onto the vectors v1 and v2, with the input being a matrix that projects on column (a1) and (b1). The person is unsure about how to approach the problem and asks for clarification on what a1* and b1* represent and if the projection is idempotent.
  • #1
ConeOfIce
13
0

Homework Statement


Consider the following two vectors v1= (cosx , sinx)(transpose) , v2= (-sinx , cosx)(transpose). Compute the projectors P1, P2 onto the vectors v1 and v2.


Homework Equations


(a1) (a1*,b1*) (A) <---input
(b1) ...(B)
This is a matrix
that projects on
column
(a1)
(b1)

The snowman (one bracket over the other) is supposed to be a vector.

The Attempt at a Solution



I am completely lost on this situation. The textbook is extremely unclear, and there doesn't seem to be any information on line for how you go about projectors. Am I supposed to replace the input vector with v1, and then multiply it by the matrix to see what I get, or am I supposed to solve for a1 and b1, which I presume would be the projector. Any feedback would be great

Sorry if this comes out ugly.
Thanks.
 
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  • #2
What are a1* and b1*? Is the projection you defined idempotent?
 

Related to Calculate projections of vectors

1. How do you calculate projections of vectors?

The projection of vector a onto vector b is given by the formula projba = (a · b / |b|2) * b. This means taking the dot product of a and b, dividing it by the square of the magnitude of b, and then multiplying it by b.

2. What is the purpose of calculating projections of vectors?

Calculating projections of vectors is useful in many areas of science, such as physics, engineering, and mathematics. It can help in visualizing and understanding the relationship between two vectors, and it is also used in solving problems involving forces, motion, and geometry.

3. Can you give an example of calculating projections of vectors?

Sure, let's say we have two vectors a = (3, 4) and b = (1, 2). To calculate the projection of a onto b, we first find the dot product: a · b = (3*1) + (4*2) = 11. Then, we find the magnitude of b: |b| = √(12 + 22) = √5. Plugging these values into the formula, we get projba = (11 / 5) * (1, 2) = (11/5, 22/5) ≈ (2.2, 4.4).

4. How do you interpret the result of a calculated projection?

The result of a calculated projection is a vector that lies on the same line as the vector being projected onto. In the example above, the result (2.2, 4.4) is a multiple of b = (1, 2), which means it is parallel to b. Additionally, the magnitude of the result is equal to the length of the component of a in the direction of b.

5. Are there any special cases to consider when calculating projections of vectors?

Yes, there are two main cases to consider when calculating projections of vectors. The first is when the two vectors are perpendicular to each other, in which case the projection will be zero since the dot product will be zero. The second case is when the vector being projected onto has a magnitude of zero, in which case the projection is undefined.

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