Vector addition

Question

I am studying vectors on my own and am struggling with this question.

I have tried sketching diagrams but cannot visualise the situation or solve the problem.

Given that HL + KN = KL + HM show that the points M and N are coincident.

Answer 1

If $\vec{HL}+\vec{KN}=\vec{KL}+\vec{HM}$, then:

$$\begin{array}{rl}\vec{HM}&=\vec{HL}+\vec{KN}-\vec{KL}\\&=\vec{HL}+\vec{LK}+\vec{KN}\\&=\vec{HN}\end{array}$$

and from $\vec{HM}=\vec{HN}$ you conclude $M=N$.

Answer 2

Put the origin $O$ in your diagram, and remember that $$\overrightarrow{HL} = \overrightarrow{OL}-\overrightarrow{OH}.$$ Now proceed.

Answer 3

Rearranging the equation HL + KN = KL + HM:

HL + KN - KL = HM

Notice that -KL = LK, so we can say:

HL + KN + LK = HM

Now look at the diagram below:

In the diagram, we can deduce that HL + KN + LK = HN.

Since we know that HL + KN + LK = HM from the equation you provided, we know that HM = HN. Thus, M and N must be the same point.