Hey there - I'm stumped with this piece of revision. Can't find many if any examples in my class notes. Can anyone offer some comments on my attempt and how I can complete this - Hopefully I can learn and use this as an example as ther are others.

The position vectors of point A,B & C, relative to a fixed origin (O), are:

a = 6i + 4j - k

b = 8i + 5j - 3k

c = 2i + 8j - 5k

(i)Find the vectorAB, the lenghtAB, the cosine of angle ABC and the area of triangle ABC

(ii) Find the vector equation of the straight line passing through A and B

(iii) Find the co-ordss of the point D on the line for whichODis prependicular toAB. Hence, or otherwise calculate the shortest distance from O to the lineAB

OK, so for (i)

AB = b - a

AB = 2i + j - 2k

Modulus of AB is 3

To find the cosine of angle ABC I need the vector and modulus of CB,

which is b -c = 6i - 3j +2k and mod. = 7

The anlge is cos x = (AB * CB)/(mod. AB * mod. CB)

cos x = 5/21

For the area: The length/modulus of AC and CB multiplied and divided by 2

mod. AC = 4*sqrt3

mod. CB = 7

Area therefore, 14sqrt3

(ii) vector equation of line passing through A, B

let P be any point on the line with vector r.

OP = OA + AP

AP = t*AB

AP = t*(b-a)

AB = (b - a) = 2i + j - 2k

AP = 2ti + tj - 2tk

therfore OP = OA + AP

OP = 6i + 4j - k + 2ti + tj - 2tk

OP = (6 +2t)i + (4+t)j - (1 + 2t)k

(iii) no idea.

How have I gotten on with my parts (i) & (ii)?

Thanks, D