Vectors & Matrices

10 methods · 22 questions

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Any point on the line through $A$ and $B$ is $r = (1 - α) a + α b$ , and the single condition that the weights sum to $1$ is exactly the test for collinearity. Read geometry straight off the weights: $α$ is the fraction of the way from $A$ to $B$ , and a point dividing $A B$ in ratio $p : q$ is $\frac{q a + p b}{p + q}$ .

Trigger: a point is described by a ratio on a segment, lies on a join of two named points, or you must prove three points are collinear.

Instances:
(i) Place a point dividing $A B$ in a given ratio, then impose a further condition (angle bisector, equal lengths) to pin the parameter.
(ii) Prove collinearity or find where a constructed line cuts a side by forcing the three weights of a point to sum to $1$ .
(iii) Write a point as a weighted average of three vertices, $r = α a + β b + γ c$ with $α + β + γ = 1$ , to land it in a plane or on a cevian, and recover side-ratios from the weights.

Linked questions (3)

STEP 2 2006 · Q8

Show that the line through the points with position vectors

x

and

y

has equation

[[B]] r [[/ B]] = (1 - α) [[B]] x [[/ B]] + α [[B]] y [[/ B]],

where

α

is a scalar parameter.

The sides

O A

and

C B

of a trapezium

O A B C

are parallel, and

O A > C B

. The point

E

O A

is such that

O E : E A = 1 : 2

, and

F

is the midpoint of

C B

. The point

D

is the intersection of

O C

produced and

A B

produced; the point

G

is the intersection of

O B

and

E F

; and the point

H

is the intersection of

D G

produced and

O A

. Let

a

and

c

be the position vectors of the points

A

and

C

, respectively, with respect to the origin

O

Show that $B$ has position vector $λ < s t r o n g > a < / s t r o n g > + < s t r o n g > c < / s t r o n g >$ for some scalar parameter $λ$ .
Find, in terms of $a$ , $c$ and $λ$ only, the position vectors of $D$ , $E$ , $F$ , $G$ and $H$ . Determine the ratio $O H : H A$ .

STEP 2 2008 · Q8

The points

A

and

B

have position vectors

a

and

b

, respectively, relative to the origin

O

. The points

A

B

and

O

are not collinear. The point

P

lies on

A B

between

A

and

B

such that

A P : P B = (1 - λ) : λ .

Write down the position vector of

P

in terms of

a

b

and

λ

. Given that

O P

bisects

∠ A O B

, determine

λ

in terms of

a

and

b

, where

a = ∣ a∣

and

b = ∣ b ∣

.

The point

Q

also lies on

A B

between

A

and

B

, and is such that

A P = B Q

. Prove that

O Q^{2} - O P^{2} = (b - a)^{2} .

STEP 2 2009 · Q8

The non-collinear points

A

B

and

C

have position vectors

a

b

and

c

, respectively. The points

P

and

Q

have position vectors

p

and

q

, respectively, given by

[[B]] p [[/ B]] = λ [[B]] a [[/ B]] + (1 - λ) [[B]] b [[/ B]] and [[B]] q [[/ B]] = μ [[B]] a [[/ B]] + (1 - μ) [[B]] c [[/ B]]

where

0 < λ < 1

and

μ > 1

. Draw a diagram showing

A

B

C

P

and

Q

.

Given that

C Q \times B P = A B \times A C

, find

μ

in terms of

λ

, and show that, for all values of

λ

, the the line

P Q

passes through the fixed point

D

, with position vector

[[B]] d [[/ B]]

given by

[[B]] d = - a + b + c [[/ B]]

. What can be said about the quadrilateral

A B D C