If Vector a = (5, -5, 3), Vector b = (1, 2, 3), determin proj (Vector a onto Vector b).

(Full solutions)

To determine the projection of vector a onto vector b, we need to use the formula for the projection of a vector onto another vector:

proj(a, b) = (a · b) / ||b||^2 * b

where a · b is the dot product of vectors a and b, ||b|| is the magnitude (or length) of vector b, and * denotes scalar multiplication.

Let's compute the projection step by step.

Step 1: Calculate the dot product of vectors a and b.
The dot product of two vectors a and b is calculated by multiplying their corresponding components and summing them.

a · b = (5 * 1) + (-5 * 2) + (3 * 3)
= 5 - 10 + 9
= 4

Step 2: Calculate the magnitude squared of vector b.
The magnitude of a vector b can be calculated using the formula:

||b||^2 = b · b

||b||^2 = (1 * 1) + (2 * 2) + (3 * 3)
= 1 + 4 + 9
= 14

Step 3: Compute the projection of vector a onto vector b.
Using the formula proj(a, b) = (a · b) / ||b||^2 * b:

proj(a, b) = (4) / (14) * (1, 2, 3)
= (4/14) * (1, 2, 3)
= (2/7) * (1, 2, 3)
= (2/7, 4/7, 6/7)

Therefore, the projection of vector a onto vector b is (2/7, 4/7, 6/7).