Q3

Let u =

(i)

(ii)

(iii)

(iv)

Q

S

T

ν

'P

-0.

U

R and w =

1

α.

Calculate the angle between the two vectors u and v.

Find the value of a such that u and w are orthogonal.

Find the value of a such that u, v, w are linearly dependent.

[5 marks]

[5 marks]

[15 marks]

State the "Basis Theorem". Choose a value of a, different from the one that you

have found in (iii), so that u, v and w become linearly independent, apply the

"Basis Theorem" to obtain a basis for the three-dimensional space R3 using

´2

Gaussian Elimination Method. Find the coordinates of 3 in terms of this basis.

5

(Marks will ONLY be awarded for Gaussian Elimination Method.)

[15 marks]