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Question
Show that the following points are collinear:
P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0).
Solution
Let `bar"p", bar"q", bar"r"` be position vectors of the points.
P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0) respectively.
Then `bar"p" = 4hati + 5hatj + 2hatk, bar"q" = 3hati + 2hatj + 4hatk, bar"r" = 5hati + 8hatj + 0hatk`
`bar("PQ") = bar"q" - bar"p"`
= `(3hati + 2hatj + 4hatk) - (4hati + 5hatj + 2hatk)`
= `-hati - 3hatj + 2hatk`
= `-(hati + 3hatj - 2hatk)` .....(1)
and `bar("QR") = bar"r" - bar"q"`
= `(5hati + 8hatj + 0hatk) - (3hati + 2hatj + 4hatk)`
= `2hati + 6hatj - 4hatk`
= `2(hati + 3hatj - 2hatk)`
= `2.bar("PQ")` ....[By(1)]
∴ `bar("QR")` is a non-zero scalar multiple of `bar("PQ")`
∴ They are parallel to each other.
But they have point Q in common.
∴ `bar("PQ")` and `bar("QR")` are collinear vectors.
Hence, the points P, Q, and R are collinear.
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