Porisma

Worked solution

Build a line by extending and splitting a displacement

The line l passes through P and Q. The position vector of P is p\mathbf p and PQ→=m\overrightarrow{PQ}=\mathbf m.

(a) Express the position vector of Q in terms of p,m.

(b) E lies on PQ produced with PE=3 PQ; F is the midpoint of PQ; G lies on QP produced with PG:PQ=2:1. (i) Mark E,F,G on the diagram. (ii) Express their position vectors in terms of p,m.

(c) What can you conclude about the position vector of any point R on l?

Expand diagram

Use the scrollbars or arrow keys to inspect details.

The source line l passes through P and Q, with origin O below it, position vector OP=p and displacement PQ=m. The positions of E,F,G are to be supplied.
The source line l passes through P and Q, with origin O below it, position vector OP=p and displacement PQ=m. The positions of E,F,G are to be supplied.
The source line l passes through P and Q, with origin O below it, position vector OP=p and displacement PQ=m. The positions of E,F,G are to be supplied.

CJC · Lecture · Exploration Activity

(a) Add the displacement from P to Q

OQ→=OP→+PQ→=p+m\overrightarrow{OQ}=\overrightarrow{OP}+\overrightarrow{PQ}=\mathbf p+\mathbf m.

How did this part go?

(b)(i) Mark the points in the correct order

E is beyond Q in the P→Q direction, F is halfway along PQ, and G is beyond P in the opposite direction. Their order along the line is G,P,F,Q,E. The completed figure marks all three positions.

How did this part go?

(b)(ii) Use the signed multiples of m

OE→=p+3m\overrightarrow{OE}=\mathbf p+3\mathbf m, OF→=p+12m\overrightarrow{OF}=\mathbf p+\tfrac12\mathbf m, and OG→=p−2m\overrightarrow{OG}=\mathbf p-2\mathbf m. The negative sign places G opposite the direction from P to Q.

How did this part go?

(c) Let the scalar range over all real values

Every point R has OR→=p+λm\overrightarrow{OR}=\mathbf p+\lambda\mathbf m for some λ∈R\lambda\in\mathbb R. Here P,Q are distinct, so m≠0. Negative values extend behind P, values between 0 and 1 lie within PQ, and values above 1 extend beyond Q.

How did this part go?

Compare your sketch

Expand diagram

Use the scrollbars or arrow keys to inspect details.

The completed source diagram places G two displacements behind P, F halfway from P to Q and E three displacements beyond P. Their position vectors are p−2m,p+½m,p+3m; Q has p+m.
The completed source diagram places G two displacements behind P, F halfway from P to Q and E three displacements beyond P. Their position vectors are p−2m,p+½m,p+3m; Q has p+m.
The completed source diagram places G two displacements behind P, F halfway from P to Q and E three displacements beyond P. Their position vectors are p−2m,p+½m,p+3m; Q has p+m.

Common mistakes

Use a negative displacement for G; a length ratio alone does not choose the direction.

Read the hint