Final answer:
The scalar projection of b onto a is 9 and the vector projection is (54/71, 63/71, -54/71).
Step-by-step explanation:
To find the scalar projection of vector b onto vector a, we use the formula
scalar projection = (b ⋅ a) / |a|
Substituting the given values, we get
scalar projection = ((3 ⋅ 6) + (-1 ⋅ 7) + (1 ⋅ -6)) / ∣(6, 7, -6)∣
Calculating the dot product and magnitude, we find that the scalar projection is 9.
To find the vector projection of b onto a, we multiply the scalar projection by the unit vector in the direction of a:
vector projection = scalar projection ⋅ (a/|a|)
Substituting the values, we get
vector projection = 9 ⋅ (6/∣(6, 7, -6)∣, 7/∣(6, 7, -6)∣, -6/∣(6, 7, -6)∣)
Simplifying the expression, we find that the vector projection is (54/71, 63/71, -54/71).