Find the projection of the vector u = [1; 2; 3] on the vector v = [1; -1; 0].
Question
Find the projection of the vector u = [1; 2; 3] on the vector v = [1; -1; 0].
Solution
The projection of a vector u onto a vector v is given by the formula:
proj_v(u) = ((u.v) / ||v||^2) * v
where "." denotes the dot product, and ||v|| denotes the magnitude of vector v.
Step 1: Calculate the dot product of u and v.
u.v = (11) + (2-1) + (3*0) = 1 - 2 + 0 = -1
Step 2: Calculate the magnitude of vector v.
||v|| = sqrt((1)^2 + (-1)^2 + (0)^2) = sqrt(1 + 1 + 0) = sqrt(2)
Step 3: Square the magnitude.
||v||^2 = (sqrt(2))^2 = 2
Step 4: Substitute the values into the formula.
proj_v(u) = ((-1) / 2) * [1; -1; 0] = [-0.5; 0.5; 0]
So, the projection of the vector u = [1; 2; 3] on the vector v = [1; -1; 0] is [-0.5; 0.5; 0].
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