Tamil Nadu Board of Secondary EducationHSC Arts Class 11

# Show that the following vectors are coplanar ijkijkjki^-2j^+3k^,-2i^+3j^-4k^,-j^+2k^ - Mathematics

Sum

Show that the following vectors are coplanar

hat"i" - 2hat"j" + 3hat"k", -2hat"i" + 3hat"j" - 4hat"k", -hat"j" + 2hat"k"

#### Solution

Let the given vectors be vec"a" = hat"i" - 2hat"j" + 3hat"k"

vec"b" = -2hat"i" + 3hat"j" - 4hat"k"

and vec"c" = -hat"j" + 2hat"k"

Three vectors vec"a", vec"b" and vec"c" are coplanar if one vector is expressed as a linear combination of the other two vectors.

vec"a" = "s"vec"b" + "t"vec"c" where s, t are scalars.

Let vec"a" = "s"vec"b" + "t"vec"c" where s, t are scalars.

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

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

1 = – 2S   ........(1)

– 2 = 3s  – t   .......(2)

3 =  – 4s + 2t

(1) ⇒ s = -1/2

Substituting for s in equation (2) we get

– 2 = 3 xx - 1/2 - "t"

– 2 = - 3/2 - "t"

t = - 3/2 + 2

= (- 3 + 4)/2

t = 1/2

Substituting for s and t in equation (3)

(3) ⇒ 3 = -4 xx -1/2 + 2 xx 1/2

3 = + 2 + 1

3 = 3

∴ Equation (3) is satisfied.

The scalars s and t exist.

∴ The vectors vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = -2hat"i" + 3hat"j" - 4hat"k" and vec"c" = -hat"j" + 2hat"k" are coplanar vectors.

Concept: Representation of a Vector and Types of Vectors
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Chapter 8: Vector Algebra - Exercise 8.2 [Page 68]

#### APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 8 Vector Algebra
Exercise 8.2 | Q 9. (i) | Page 68
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