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Question
Show that the points A (-7 , 4 , -2),B (-2 , 1 , 0)and C (3 ,-2 ,2) are collinear.
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Solution
Let `bar "a" , bar "b" , & bar"c"` be position vectors of A,B &C
`bar"AB" = bar"b" -bar"a"`
= `(-2hat"i" +hat"j") -(-7hat"i" +4 hat"j" - 2hat"k")`
=`5hat "i" -3 hat"j" +2hat"k"`
`bar"AC" = bar"c" -bar"a"`
= `(3 hat"i" - 2 hat"j" +2 hat"k") - (-7hat"i" +4 hat"j" -2hat"k")`
=`10 hat"i" -6 hat"j" +4 hat"k"`
= `2 [5hat"i" -3 hat"j" +2hat"k"]`
`=> bar"AC" = 2 bar(("AB"))`
`=> bar "AC" "is a scalar multiple of "bar "AB"`
`=> bar "AC" & bar "AB"`
∵ A is common
⇒ A B &C are collinear.
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