Let S u upsilon omega where u 3 1 1 upsilon 1 2 1 omega

Let S = {u, upsilon, omega} where u = [3 1 1] upsilon = [-1 2 1] omega = [-1/2 -2 7/2] (a) Is S ail orthogonal set? (b) Is S all orthonormal set? (c) Is S an orthogonal basis of R^3? (d) Express the vector [6 1 -8] using the vectors in S.

Solution

Calculate dot product:

u · v = u_x · v_x + u_y · v_y + u_z · v_z = 3 · (-1) + 1 · 2 + 1 · 1 = -3 + 2 + 1 = 0

Vectors are orthogonal, as their dot product is equal to zero.

Calculate dot product:

u · w = u_x · w_x + u_y · w_y + u_z · w_z = 3 · -12 + 1 · (-2) + 1 · 72 = -32 - 2 + 72 = 0

Vectors are orthogonal, as their dot product is equal to zero.

Calculate dot product:

v · w = v_x · w_x + v_y · w_y + v_z · w_z = (-1) · -12 + 2 · (-2) + 1 · 72 = 12 - 4 + 72 = 0

Vectors are orthogonal, as their dot product is equal to zero

=> dot product of u.v.w=0 +> S is Orthogonal