Calculus orthogonal

Question

Find the orthogonal projection of v = 16, 12, 18, 14

X1 = -3, 4, 2, -4

X2 = -4, -5, 4, 0

Answer 1

Let us start with X1.

What is an orthogonal projection of v onto X1? It is a vector, whose length is the absolute value of the dot product of v and X1, and which is pointed in the direction of X1 if the dot product is positive or opposite to the direction of X1, if the dot product is negative. It can be found using the formula below

v_x1 = X1 (v•X1)/(X1•X1)

1. v•X1 = 16*(-3) + 12*4 + 18*2 + 14*(-4) = - 20
2. X1•X1 = (-3)² + (4)² +(2)² +(-4)² = 9+16+4+16 = 45
3. (v•X1)/(X1•X1) = - 20/45 = - 4/9
4. v_x1 = (- 4/9) X1 = (- 4/9) (- 3, 4, 2, - 4) =

= (4/3, - 16/9, - 8/9, 16/9)

Repeat for X2.

v_x2 = X2 (v•X2)/(X2•X2)

1. v•X2 = 16*(-4) + 12*(-5) + 18*4 + 14*0 = - 52
2. X2•X2 = (-4)² + (-5)² +(4)² +(0)² = 16+20+16 = 52
3. (v•X2)/(X2•X2) = - 52/52 = - 1
4. v_x2 = (- 1) X2 = (- 1) (- 4, - 5, 4, 0) =

= (4, 5, - 4, 0)