Find the scalar and vector projection of the vector

Question

Find the scalar and vector projection of the vector b=<1,0,−1> onto the vector a=<−4,2,4>.

Scalar projection (i.e., component):
Vector projection < , , >

Answer 1

a = (-4, 2, 4) and b = (1, 0, -1)

 

a•a = (-4)(-4) + (2)(2) + (4)(4) = 36

a•b = (-4)(1) + (2)(0) + (4)(-1) =-8

b•b = (1)(1) + (0)(0) + (-1)(-1) =2

 

|a| = 6

|b| = √2

 

There exists a scalar, σ, and a vector u such that

• u is perpendicular to a
• b = σa + u
• σ|a| is the scalar projection of b onto a
• σa is the vector projection of b onto a

 

Then a•b = σ a•a + a•u ⇒  

        a•b = σ a•a ⇒

        σ =  a•b /  a•a = -8/36 = -2/9

 

The scalar projection of b onto a is σ|a| = (-2/9)(6) = -4/3

 

The vector projection of b onto a is σa = (-2/9)(-4, 2, 4) = (8/9, -4/9, -8/9)