Mathematics

Mathematics

Problem

Let

A(a,1,0)

and

B(2,3,-1)

be two points, where

a

is a real number. A point

C

divides

AB

internally in the ratio

2:3

. a. (i) Find the position vector of

C

in terms of

a

. b. (ii) Prove that the magnitude of

\overrightarrow{AC}

is

\dfrac{2}{5}

times the magnitude of

\overrightarrow{AB}

for every real

a

. c. (iii) Generalise the result: if a point

C

divides the segment joining any two points

A

and

B

internally in the ratio

m:n

, show that

|\overrightarrow{AC}| = \dfrac{m}{m+n}\, |\overrightarrow{AB}|

.

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