Question

Find the scalar and vector projections of b onto a. Given a = i j k and b = i - j k, what is the scalar projection of b onto a? What is the vector projection of b onto a?

Answer 1

The scalar projection of b onto a is -√3 / 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 need to calculate the dot product of a and b divided by the magnitude of a.

Scalar projection of b onto a:

Scalar projection = (a · b) / |a|

Given a = (i + j + k) and b = (i - j - k), the dot product of a and b is a·b = (1)(1) + (1)(-1) + (1)(-1) = 1 - 1 - 1 = -1

The magnitude of a is |a| = √(1^2 + 1^2 + 1^2) = √3

Therefore, the scalar projection of b onto a is (-1) / √3 = -√3 / 3.

Vector projection of b onto a:

Vector projection = (scalar projection) * (unit vector of a)

To find the unit vector of a, divide a by its magnitude: a/|a|

Unit vector of a = (i + j + k) / √3 = (√3/3)(i + j + k)

Multiplying the scalar projection by the unit vector of a gives the vector projection: (-√3 / 3)(√3/3)(i + j + k)

Vector projection = (-1/3)(i + j + k).