Final answer:
The scalar projection of vector b onto vector a is -1/√3, and the vector projection of b onto a is (-1/3) * (i + j + k).
Step-by-step explanation:
To find the scalar projection of vector b onto vector a, we calculate the dot product of vectors a and b (also known as the scalar product), and divide it by the magnitude of vector a. This gives us the length of the projection of b onto a. The formula for the scalar projection is given by:
scalar_proj_of_b_onto_a = (a ⋅ b) / |a|
To find the vector projection of b onto a, we multiply the scalar projection by the unit vector in the direction of a. The formula for the vector projection is given by:
vector_proj_of_b_onto_a = (a ⋅ b / |a|^2) * a
Given vectors a = i + j + k and b = i - j - k, first calculate the dot product:
a ⋅ b = (1)(1) + (1)(-1) + (1)(-1) = 1 - 1 - 1 = -1
The magnitude of vector a is |a| = √(1^2 + 1^2 + 1^2) = √3
The scalar projection of b onto a is:
scalar_proj_of_b_onto_a = (-1) / √3 = -1/√3
The vector projection of b onto a is:
vector_proj_of_b_onto_a = (-1/√3^2) * (i + j + k) = (-1/3) * (i + j + k)