###### Advertisements

###### Advertisements

Show that the following vectors are coplanar

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

###### Advertisements

#### 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.

#### APPEARS IN

#### RELATED QUESTIONS

Find the value of λ for which the vectors `vec"a" = 3hat"i" + 2hat"j" + 9hat"k"` and `hat"b" = hat"i" + lambdahat"j" + 3hat"k"` are parallel

Show that the following vectors are coplanar

`2hat"i" + 3hat"j" + hat"k", hat"i" - hat"j", 7hat"i" + 3hat"j" + 2hat"k"`

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

The position vectors `vec"a", vec"b", vec"c"` of three points satisfy the relation `2vec"a" - 7vec"b" + 5vec"c" = vec0`. Are these points collinear?

The position vectors of the points P, Q, R, S are `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2hat"j" - 3hat"k"`, and `hat"i" - 6hat"j" - hat"k"` respectively. Prove that the line PQ and RS are parallel

Find the value or values of m for which `"m"(hat"i" + hat"j" + hat"k")` is a unit vector

Choose the correct alternative:

If `vec"a" + 2vec"b"` and `3vec"a" + "m"vec"b"` are parallel, then the value of m is

Choose the correct alternative:

If ABCD is a parallelogram, then `vec"AB" + vec"AD" + vec"CB" + vec"CD"` is equal to

Choose the correct alternative:

One of the diagonals of parallelogram ABCD with `vec"a"` and `vec"b"` as adjacent sides is `vec"a" + vec"b"`. The other diagonal `vec"BD"` is

Choose the correct alternative:

If `vec"a", vec"b", vec"c"` are the position vectors of three collinear points, then which of the following is true?

Choose the correct alternative:

If the points whose position vectors `10hat"i" + 3hat"j", 12hat"i" - 5hat"j"` and `"a"hat"i" + 11hat"j"` are collinear then a is equal to