If Vector a = (5, -5, 3), Vector b = (1, 2, 3), determin proj (Vector a onto Vector b).
(Full solutions)
done, see your earlier post
To find the projection of vector a onto vector b, you need to use the formula for vector projection:
proj(a onto b) = ((a · b) / ||b||^2) * b
where a · b is the dot product of vectors a and b, ||b||^2 is the magnitude squared of vector b, and * denotes scalar multiplication.
Let's calculate it step by step:
Step 1: Calculate the dot product of vectors a and b
a · b = (5 * 1) + (-5 * 2) + (3 * 3)
= 5 - 10 + 9
= 4
Step 2: Calculate the magnitude squared of vector b
||b||^2 = (1^2) + (2^2) + (3^2)
= 1 + 4 + 9
= 14
Step 3: Calculate the scalar multiplication
((a · b) / ||b||^2) = (4 / 14) ≈ 0.2857
Step 4: Calculate the projection of vector a onto vector b
proj(a onto b) = ((a · b) / ||b||^2) * b
= 0.2857 * (1, 2, 3)
≈ (0.286, 0.571, 0.857)
Therefore, the projection of vector a onto vector b is approximately (0.286, 0.571, 0.857).