Tamil Nadu Board of Secondary EducationHSC Arts Class 11th

# Show that the points whose position vectors ijkjkijk4i^+5j^-k^,-j^-k^,3i^+9j^+4k^ and ijk-4i^+4j^+4k^ are coplanar - Mathematics

Sum

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.

Concept: Representation of a Vector and Types of Vectors
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