Karnataka Board PUCPUC Science 2nd PUC Class 12

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar. - Mathematics

Advertisements
Advertisements

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

Advertisements

Solution

Position vector of `vec(OA)=4hati+hatj+2hatk`

Position vector of `vec(OB)=5hati+xhatj+6hatk`

Position vector of `vec(OC)=5hati+hatj-hatk`

Position vector of `vec(OD)=7hati+4hatj+0hatk`

`vec(AB)=vec(OB)-vec(OA)`

`=5hati+xhatj+6hatk-4hati-hatj-2hatk=hati+(x-1)hatj+4hatk`

`vec(AC)=vec(OC)-vec(OA)`

`=5hati+hatj-hatk-4hati-hatj-2hatk=hati-3hatk`

`vec(AD)=vec(OD)-vec(OA)`

`=7hati+4hatj+0hatk-4hati-hatj-2hatk=3hati+3hatj-2hatk`

The above three vectors are coplanar

`=>vec(AB).(vec(AC)xxvec(AD))=0`

`=>|[1,x-1,4],[1,0,-3],[3,3,-2]|=0`

`=>1(0 + 9) - (x - 1)( -2 + 9) + 4(3 - 0) = 0`

`=>9-7(x-1)+12=0`

`=>-7(x-1)=-21`

`=>x-1=3`

`therefore x=4`

 

Concept: Vectors Examples and Solutions
  Is there an error in this question or solution?
2014-2015 (March) Panchkula Set 1

RELATED QUESTIONS

If a unit vector `veca` makes angles `pi/3` with `hati,pi/4` with `hatj` and acute angles θ with ` hatk,` then find the value of θ.


 

If `veca=2hati+hatj+3hatk and vecb=3hati+5hatj-2hatk` ,then find ` |veca xx vecb|`

 

 

A line passing through the point A with position vector `veca=4hati+2hatj+2hatk` is parallel to the vector `vecb=2hati+3hatj+6hatk` . Find the length of the perpendicular drawn on this line from a point P with vector `vecr_1=hati+2hatj+3hatk`

 

if `|vecaxxvecb|^2+|veca.vecb|^2=400 ` and `|vec a| = 5` , then write the value of `|vecb|`


If `vecr=xhati+yhatj+zhatk` ,find `(vecrxxhati).(vecrxxhatj)+xy`


Find `veca.(vecbxxvecc), " if " veca=2hati+hatj+3hatk, vecb=-hati+2hatj+hatk  " and " vecc=3hati+hatj+2hatk`


Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).


Find the angle between the vectors `vec"a" + vec"b" and  vec"a" -vec"b" if  vec"a" = 2hat"i"-hat"j"+3hat"k" and vec"b" = 3hat"i" + hat"j"-2hat"k", and"hence find a vector perpendicular to both"  vec"a" + vec"b" and vec"a" - vec"b"`.


If `vec"a" + vec"b" + vec"c"` = 0, show that `vec"a" xx vec"b" = vec"b" xx vec"c" = vec"c" xx vec"a"`. Interpret the result geometrically?


Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).


Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.


The vectors from origin to the points A and B are `vec"a" = 2hat"i" - 3hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + 3hat"j" + hat"k"`, respectively, then the area of triangle OAB is ______.


For any vector `vec"a"`, the value of `(vec"a" xx hat"i")^2 + (vec"a" xx hat"j")^2 + (vec"a" xx hat"k")^2` is equal to ______.


If `|vec"a"|` = 10, `|vec"b"|` = 2 and `vec"a".vec"b"` = 12, then value of `|vec"a" xx vec"b"|` is ______.


The vectors `lambdahat"i" + hat"j" + 2hat"k", hat"i" + lambdahat"j" - hat"k"` and `2hat"i" - hat"j" + lambdahat"k"` are coplanar if ______.


If `|vec"a"|` = 4 and −3 ≤ λ ≤ 2, then the range of `|lambdavec"a"|` is ______.


The value of the expression `|vec"a" xx vec"b"|^2 + (vec"a".vec"b")^2` is ______.


Share
Notifications



      Forgot password?
Use app×