vector and scalars

So many questions......Hope that the hints below can help you. I appreciate your effort in doing maths exercises like this.

1. Let AN : NB = 1 : r. Using section formula to find ON in terms of vectors a and b.
Use (a) to express your answer in terms of a and c.
As ON // OC, we have CN = kOC for some real number k. Now by comparing coefficients, you can evaluate r.

2. (a) Using similar triangles, you can find DE : EN. Now use section formula.
(b) Similarly, you can find FC. By finding AC, you can complete the proof.

3. Of course you can use this property. The rest is just similar to your question 1(a).

4. Yes. But if you eliminate k like this, you need to make sure that r, q =/= 0 as they appear in the denominator. One better method is as follows:

q+kq-s=0........(1)
p+kr=0.............. .(2)

(1) * r - (2) * q,

qr + kqr - sr - pq - kpq = 0
qr - sr - pq = 0
qr = pq + rs

This avoids dividing the question into different cases.

scalars quantities have magnitude

vector quantities have magnitude and directions