If Vector a = (5, -5, 3), Vector b = (1, 2, 3), determin proj (Vector a onto Vector b).
that would be
a•b/|b| =
(5)(1)+(-5)(2)+(3)(3)
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√(1^2+2^2+3^2)
To determine the projection of Vector a onto Vector b, we can use the formula for projecting one vector onto another. The formula is:
proj(a onto b) = (a dot b) / ||b||^2 * b
Let's break this formula down step by step:
Step 1: Calculate the dot product of Vector a and Vector b.
The dot product of two vectors is found by multiplying the corresponding components together and then summing them up. So, in this case, we have:
a dot b = (5 * 1) + (-5 * 2) + (3 * 3) = 5 - 10 + 9 = 4
Step 2: Calculate the magnitude (length) of Vector b.
The magnitude or length of a vector is found using the formula:
||b|| = √(b1^2 + b2^2 + b3^2)
For Vector b, we have:
||b|| = √(1^2 + 2^2 + 3^2) = √(1 + 4 + 9) = √14
Step 3: Calculate the square of the magnitude of Vector b.
To simplify the formula, we need to square the magnitude of Vector b. So we have:
||b||^2 = (√14)^2 = 14
Step 4: Calculate the projection of Vector a onto Vector b.
Using the formula mentioned earlier, we can now determine the projection:
proj(a onto b) = (a dot b) / ||b||^2 * b
= 4 / 14 * (1, 2, 3)
= (4/14 * 1, 4/14 * 2, 4/14 * 3)
= (4/14, 8/14, 12/14)
= (2/7, 4/7, 6/7)
Therefore, the projection of Vector a onto Vector b is (2/7, 4/7, 6/7).