Question

Show that the vectors a = 3i − j + 4k, b = i − 3j − 2k and c = 4i − 3j + 2k are
linearly independent. Find numbers α, β and γ such that d = 2i + 3j − k can be
expressed in the form d = αa + βb + γc

Answer 1

Step-by-step explanation:

Consider: p(3, -1, 4) + q(1, -3, -2) + r(4, -3, 2) = 0

So: 3p + q + 4r = 0, -p - 3q - 3r = 0, 4p - 2q + 2r = 0

Try solving: if you get p = q = r = 0, the vectors are linearly independent

Now: (2, 3, -1) = s(3, -1, 4) + t(1, -3, -2) + u(4, -3, 2)

So: 2 = 3s + t + 4u, 3 = -s - 3t - 3u, -1 = 4s - 2t + 2u

Solve for s, t, u