Points , and are given in space, where is a real…

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Points

A(1,2,-1)

B(4,8,k)

and

C(7,14,5)

are given in space, where

k

is a real number. (i) Write the position vectors

\vec{OA}

\vec{OB}

and

\vec{OC}

in component form, taking

O

as the origin. (ii) Obtain the vectors

\vec{AB}

and

\vec{AC}

. (iii) Determine the value of

k

for which

A,\,B

and

C

are collinear. (iv) For this value of

k

, find the ratio in which

B

divides

AC

internally.