To find the signed magnitude c of the projection of x1 onto x2, we first need to find the unit vector u in the direction of the projection.
The projection of x1 onto x2 can be calculated using the formula:
projx1x2 = (x1 · x2 / ||x2||^2) * x2
Where · denotes the dot product and ||x2||^2 is the squared magnitude of x2.
Since x1 and x2 are given as vectors, we can calculate their dot product by multiplying their corresponding components and summing the results:
x1 · x2 = a1*a1 + a2*(-a2) + a3*a3
= a1^2 - a2^2 + a3^2
The magnitude of x2 can be calculated using the formula:
||x2|| = √(a1^2 + (-a2)^2 + a3^2)
= √(a1^2 + a2^2 + a3^2)
Now, we can calculate c using the formula:
c = (x1 · x2) / ||x2||^2
c = (a1^2 - a2^2 + a3^2) / (a1^2 + a2^2 + a3^2)
Therefore, the signed magnitude c of the projection of x1 onto x2 is (a1^2 - a2^2 + a3^2) / (a1^2 + a2^2 + a3^2), and the unit vector u in the direction of the projection is given by u = [a1 -a2 a3] / ||x2||.
What is the signed magnitude c of the projection of x1 onto x2? More precisely, let u be the unit vector in the direction of the correct choice above, find a number c such that the projection of x1 onto x2 = cu.
x1=[a1 a2 a3]
x2 = [a1 -a2 a3]
1 answer