Porisma

Worked solution

Distinguish internal division from a line produced

Given that a\mathbf a and b\mathbf b are the position vectors of AA and BB respectively relative to origin OO, find, in terms of a,b\mathbf a,\mathbf b, the position vectors of: (i) PP which divides ABAB in the ratio 3:73:7; (ii) RR on BABA produced such that AR/BR=2/7AR/BR=2/7; (iii) QQ on ABAB produced such that AQ/BQ=7/3AQ/BQ=7/3.

CJC · Vectors Basic · Example 5

(i) Internal division

Answer

OP→=710a+310b\overrightarrow{OP}=\tfrac7{10}\mathbf a+\tfrac3{10}\mathbf b.

Place P between A and B

OP→=7a+3b7+3=710a+310b\begin{aligned}&\overrightarrow{OP}\\ & =\frac{7\mathbf a+3\mathbf b}{7+3}\\ & =\frac7{10}\mathbf a+\frac3{10}\mathbf b\end{aligned}

AP:PB=3:7. The opposite segment supplies each position-vector weight.

How did this part go?

(ii) BA produced

Answer

OR→=75a−25b\overrightarrow{OR}=\tfrac75\mathbf a-\tfrac25\mathbf b.

Read the order B, A, R

a=5OR→+2b5+2⟹OR→=75a−25b\mathbf a=\frac{5\overrightarrow{OR}+2\mathbf b}{5+2}\quad\Longrightarrow\quad\overrightarrow{OR}=\frac75\mathbf a-\frac25\mathbf b

BA produced continues from B through A and beyond A. With BR=7 and AR=2 ratio units, BA=5. Apply internal division to A on BR.

How did this part go?

(iii) AB produced

Answer

OQ→=74b−34a\overrightarrow{OQ}=\tfrac74\mathbf b-\tfrac34\mathbf a.

Read the order A, B, Q

b=4OQ→+3a4+3⟹OQ→=74b−34a\mathbf b=\frac{4\overrightarrow{OQ}+3\mathbf a}{4+3}\quad\Longrightarrow\quad\overrightarrow{OQ}=\frac74\mathbf b-\frac34\mathbf a

AB produced continues from A through B and beyond B. AQ=7 and BQ=3 imply AB=4 ratio units. Now B lies internally on AQ.

How did this part go?

Common mistakes

“AB produced” and “BA produced” extend beyond different endpoints. The Teacher asks the lecturer to clarify this distinction; choose the point order before applying weights.

Solution notes

The Teacher labels its three solutions (a)–(c); the printed asks use (i)–(iii).