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]
To find the signed magnitude c of the projection of x1 onto x2, we need to calculate the dot product of x1 and x2, and then divide it by the magnitude of x2.
First, let's calculate the dot product of x1 and x2:
x1 · x2 = (a1 * a1) + (a2 * -a2) + (a3 * a3)
= a1^2 - a2^2 + a3^2
Next, let's calculate the magnitude of x2:
||x2|| = sqrt((a1)^2 + (-a2)^2 + (a3)^2)
= sqrt(a1^2 + a2^2 + a3^2)
Finally, we divide the dot product by the magnitude of x2 to find the value of c:
c = (a1^2 - a2^2 + a3^2) / sqrt(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) / sqrt(a1^2 + a2^2 + a3^2).
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||.