Answer to Question #115204 in Analytic Geometry for Bradley du Buy

Question

Let u = (4,2,-1), v = (3, 1, 1) and w = (0, 2, 1). Compute the following:
(i) 2v - 3w -u
(ii) u(w+v)
(iii) ||u . w ||
(iv) the orthogonal projection of u on w
(v) the vector component of u orthogonal to w

Answer 1

i) 2v -3w -u = 2(3,1,1) -3(0,2,1) -(4,2,-1)

(2,-6,0)

ii) u(w+v) = (4,2,-1)(0+3,2+1,1+1)

(4,2,-1)(3,3,2)

(4.3,2.3,-1.2)

(12,6,-2)

iii)||u.w||= || ( (4,2,-1).(0,2,1) ||

|| (0,4,-1) || = $\sqrt{\smash[b]{0+4.4+(-1)(-1)}}$ = $\sqrt{\smash[b]{17}}$

iv) orthogonal projection of u on w = w₁=(u.w/w²)w

[{(4,2,-1).(0,2,1)}/ (0+2.2+1.1)][ (0,2,1)]

[(0+4-1)/5][(0,2,1)

3/5(0,2,1)

(0,6/5,3/5)

v) vector component of u orthogonal to w = u-w₁

= (4,2,-1) - (0,6/5,3/5)

(4, 4/5 , -8/5 )