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Question
Show that the points whose position vectors `4hat"i" + 5hat"j" - hat"k", - hat"j" - hat"k", 3hat"i" + 9hat"j" + 4hat"k"` and `-4hat"i" + 4hat"j" + 4hat"k"` are coplanar
Solution
Let the given position vectors of the points A, B, C, D be
`vec"OA" = 4hat"i" + 5hat"j" - hat"k"`
`vec"OB" = - hat"j" - hat"k"`
`vec"OC" = 3hat"i" + 9hat"j" + 4hat"k"`
`vec"OD" = -4hat"i" + 4hat"j" + 4hat"k"`
`vec"AB" = vec"OB" - vec"OA"`
= `(-hat"j" - hat"k") - (4hat"i" + 5hat"j" + hat"k")`
= `-hat"j" - hat"k" - 4hat"i" - 5hat"j" - hat"k"`
`vec"AB" = -4hat"i" - 6hat"j" - 2hat"k"`
`vec"BC" = vec"OC" - vec"OB"`
= `(3hat"i" + 9hat"j" + 4hat"k") - (-hat"j" - hat"k")`
= `3hat"i" + 9hat"j" + 4hat"k" + hat"j" + hat"k"`
`vec"BC" = 3hat"j" + 10hat"j" + 5hat"k"`
`vec"CD" = vec"OD" - vec"OC"`
= `(-4hat"i" + 4hat"j" + 4hat"k") - (3hat"i" + 9hat"j" + 4hat"k")`
= `-4hat"i" + 4hat"j" + 4hat"k" - 3hat"i" - 9hat"j" - 4hat"k"`
`vec"CD" = -7hat"i" - 5hat"j"`
To prove the point A, B, C, D to be coplanar, it is enough to prove the vectors `vec"AB", vec"BC", vec"CD"` are coplanar.
To prove `vec"AB", vec"BC", vec"CD"` are coplanar
It is enough to prove `vec"AB" = "s"vec"BC" + "t"vec"CD"`
Three vectors `vec"AB", vec"BC", vec"CD"` are coplanar
If one vector is written as a linear combination of other two vectors.
`vec"AB" = "s"vec"BC" - "t"vec"CD"`.
∴ `(-4hat"i" - 6hat"j" - 2hat"k") = "s"(3hat"i" + 10hat"j" + 5hat"k") + "t"(-7hat"i" - 5hat"j")`
`-4hat"i" - 6hat"j" - 2hat"k" = (3"s" - 7"t")hat"i" + (10"s" - 5"t")hat"j" + 5"s"hat"k"`
Equating the like terms on both sides
– 4 = 3s – 7t ........(1)
– 6 = 10s – 5t ........(2)
– 2 = 5s ........(3)
(3) ⇒ s = `- 2/5`
Substituting in equation (2), we have
– 6 = 10 × `-2/5 - 5"t"`
– 6 = – 4 – 5t
– 6 + 4 = – 5t
⇒ – 5t = – 2
⇒ t = `2/5`
Substituting for s and t in equation (1), we have
(1) ⇒ – 4 = `3 xx -2/5 - 7 xx 2/5`
– 4 = `(-6)/5 - 14/5`
– 4 = `(-20)/5`
– 4 = – 4
∴ The sclars s and t exist.
`vec"AB" = "s"vec"BC" + "t"vec"CD"`
Hence `vec"AB", vec"BC", vec"CD"` are coplanar.
Therefore, the points A, B, C, D are coplanar points.
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