# Show that following points are collinear P(4, 5, 2), Q(3, 2, 4), R(5, 8, 0) - Mathematics and Statistics

Sum

Show that 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 the position vectors of points P, Q, R respectively.

∴ bar("p") = 4hat"i" + 5hat"j" + 2hat"k",

bar("q") = 3hat"i" + 2hat"j" + 4hat"k",

bar("r") = 5hat"i" + 8hat"j"

∴ bar("PQ") = bar("q") - bar("p")

= (3hat"i" + 2hat"j" + 4hat"k") - (4hat"i" + 5hat"j" + 2hat"k")

∴ bar("PQ") = hat"i" - 3hat"j" + 2hat"k"

bar("QR") = bar("r") - bar("q")

= 5hat"i" + 8hat"j" - (3hat"i" + 2hat"j" + 4hat"k")

= 2hat"i" + 6hat"j" - 4hat"k"

∴ bar("QR") = (-2) (-hat"i" - 3hat"j" + 2hat"k")   .......(ii)

∴ bar("QR") = (-2)  bar("PQ")   .......[From (i) and (ii)]

∴ bar("QR") is a scalar multiple of bar("PQ")

∴ bar("QR") and bar("PQ") are parallel to each other with point Q in common.

∴ bar("PQ") and bar("QR") lie on the same line.

∴ Points P, Q and R are collinear.

Concept: Vector Joining Two Points
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