Mathematics

Mathematics

Problem

Three points are given by

A(1,\,2,\,-1)

,

B(4,\,8,\,5)

and

C(7,\,14,\,k)

where

k

is a real parameter. a. (i) Find the value of

k

for which the vectors

\overrightarrow{AB}

and

\overrightarrow{AC}

are parallel. b. (ii) For the value of

k

obtained in part (i), prove that the points

A,\,B

and

C

are collinear and specify whether

B

lies between

A

and

C

. c. (iii) Determine the ratio in which point

B

divides the line segment

AC

. (Express the ratio as

AB:BC

.)

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