Suppose u 123 v 234 and w 102 a Findu2v3w b Write the vec

Solution

u = (1,2,3), v = (2,3,4), and w = (1,0,2).

(a) Find u2v+3w.

= ( 1 , 2 , 3) - 2( 2 , -3 , -4) +3(1 ,0 , -2)

= ( 1 -4 +3 , 2 -6 +0 , 3 +8 -6)

= ( 0 , -4 , 5)

b) vector (3, 4, 5) as a linear combination of u, v, and w.

c1(1, 2 , 3) +c2( 2 , -3 , -4) +c3(1 , 0 , -2) = (3 , 4 , 5)

c1 + 2c2 +c3 = 3 ------(1)

2c1 -3c2 = 4 ------(2)

3c1 -4c2 -2c3 = 5 ------(3)

Solve equations 1 , 2 , 3 :

c1 = 31/17 ; c2 = -2/17 ; c3 = 8/17

(3, 4 , 5) = 31u/17 -2v/17 + 8w/17