The parallelepiped has OA, OB and OC as three edges and remaining vertices X, Y, Z and D as shown in the diagram.
OA = 5i
OB = i + 3k
and OC = i + 4j
a) Write down the position vectors of X, Y, Z and D in terms of i, j and k and calculate the length of OD and OY.
I've done this part, but I included it so you know I have these answers if they're needed.
b) Calculate the size of angle OZY.
I know I need to use the formula u.v = |u||v|cosθ, but what vectors do I use for u and v? I tried using OZ and OY, and OZ and ZY, but neither gave me the correct answer. However, if I subtracted my "OZ and ZY" answer from 180, I got the answer in the book, but I would never have done this without already knowing the answer.
c) The point P divides CZ in the ration $\lambda : 1$, i.e. CP:PZ = $\lambda : 1$.
i. Give the position vector of P.
The problem here lies with my confusion surrounding the ratio statement. I'm new to vectors and have rarely used such ratios in the past.
ii. Find $\lambda$ if OP is perpendicular to CZ.
I should be able to do this after doing part i.