How can I answer vector on the triangle

Question

How can I get the vector OP to be this form :

I think that

• OP = b + t
• OP = a + s

I do not know how to do this problem, can anyone gives any idea and some steps to solve this ?

Answer 1

Using your definition of s and t, from triangle OAN we have

$$\vec{OP} = (1 - s)\vec{OA} + s\vec{ON} = (1 - s)\vec{a} + \frac{1}{3}s\vec{b}$$

Hence A = 1, B = 1 and C = 3.

Also from triangle OBM we have

$$\vec{OP} = t\vec{OM} + (1 - t)\vec{OB} = \frac{3}{4}t\vec{a} + (1 - t)\vec{b}$$

Hence D = 3, E = 4 and F = 1.

By comparing these 2 equations, we have

$1 - s = \frac{3}{4}t$ and $\frac{s}{3} = 1 - t$

The solution is $s = \frac{1}{3}$ and $t = \frac{8}{9}$

Therefore $\vec{OP} = \frac{2}{3}\vec{a} + \frac{1}{9}\vec{b}$