Question

show that the points A - 11 b 5,7 and c( 7,9 )all collinear​

Answer 1

Answer:

A=(1,−2,−8)

B=(5,0,−2)

C=(11,3,7)

A,B,C are collinear if

∣

∣

∣

∣

AB

∣

∣

∣

∣

+

∣

∣

∣

∣

BC

∣

∣

∣

∣

=

∣

∣

∣

∣

AC

∣

∣

∣

∣

AB

=(5−1)

i

^

+(0+2)

j

^

+(−2+8)

k

^

=4

i

^

+2

j

^

+6

k

^

∣

∣

∣

∣

AB

∣

∣

∣

∣

=

16+4+36

=2

14

BC

=(11−5)

i

^

+(3−0)

j

^

+(7+2)

k

^

=6

i

^

+3

j

^

+9

k

^

∣

∣

∣

∣

BC

∣

∣

∣

∣

=

36+9+81

=3

14

AC

=(11−1)

i

^

+(3+2)

j

^

+(7+8)

k

^

=10

i

^

+5

j

^

+15

k

^

∣

∣

∣

∣

AC

∣

∣

∣

∣

=

100+25+225

=5

14

Thus

∣

∣

∣

∣

AB

∣

∣

∣

∣

+

∣

∣

∣

∣

BC

∣

∣

∣

∣

=

∣

∣

∣

∣

AC

∣

∣

∣

∣

let B divide AC in k:1

then

(5,0,−2)=

k+1

k(11,3,7)+1(1,−2,−8)

=

k+1

(11k+1,3k−2,7k−8)

⇒k=2/3⇒k:1⇒2:3

⇒ B divide AC in 2:3.

solution

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