Home
Class 12
MATHS
The position vectors of the vertices A,B...

The position vectors of the vertices A,B and C of a triangle are `hati-hatj-3hatk,2hati+hatj-2hatk` and `-5hati+2hatj-6hatk`, respectively. The length of the bisector AD of the `angleBAC`, where D is on the segment BC, is

A

`(3)/(4)sqrt(3)`

B

`(1)/(4)`

C

`(11)/(2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A

`|AB|=|(2hati+hatj-2hatk)-(hati-hatj-3hatk)|`
`=|hati+2hatj+hatk|`
`=sqrt(1^(2)+2^(2)+1^(2))=sqrt(6)`
`|AC|=|(-5hati+2hatj-6hatk)-(hati-hatj-3hatk)|`
`=|-6hati+3hatj-3hatk|`
`=sqrt((6-6)^(2)+3^(2)+(-3)^(2))=sqrt(54)=3sqrt(6)`
`BD:DC=AB:AC=(sqrt(6))/(3sqrt(6))=(1)/(3)`
`therefore`Position vector of `D=(1(-5hati+2hatj-6hatk)+3(2hati+hatj-2hatk))/(1+3)`
`=(1)/(4)(hati+5hatj-12hatk)`
`thereforeAD=`Position vector of D-position vector of A
`AD=(1)/(4)(hati+5hatj-12hatk)-(hati-hatj-3hatk)=(1)/(4)(-3hati+9hatj)`
`=(3)/(4)(-hati+3hatj) `
`|AD|=(3)/(4)sqrt((-1)^(2)+3^(2))=(3)/(4)sqrt(10)`
Promotional Banner

Topper's Solved these Questions

  • VECTOR ALGEBRA

    ARIHANT MATHS|Exercise Exercise For Session 1|7 Videos

  • VECTOR ALGEBRA

    ARIHANT MATHS|Exercise Exercise For Session 2|17 Videos

  • TRIGONOMETRIC FUNCTIONS AND IDENTITIES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|18 Videos

Similar Questions

Explore conceptually related problems

If the position vectors of the vertices A,B and C of a DeltaABC are 7hatj+10k,-hati+6hatj+6hatk and -4hati+9hatj+6hatk , respectively, the triangle is

The position vectors of the points A, B, C are 2 hati + hatj - hatk , 3 hati - 2 hatj + hatk and hati + 4hatj - 3 hatk respectively . These points

The position vectors of the vertices of /_\ABC are : 3hati-4hatj-4hatk,2hati-hatj+hatk and hati-3hatj-5hatk respectively. Find vec(AB),vec(BC) and vec(CA) .

The position vectors of the points A,B and C are hati+2hatj-hatk,hati+hatj+hatk and 2hati+3hatj+2hatk , respectively. If A is chosen as the origin, then the position vectors of B and C are

If a=3hati-2hatj+hatk,b=2hati-4hatj-3hatk and c=-hati+2hatj+2hatk , then a+b+c is

Show that the vectors hati-hatj-hatk,2hati+3hatj+hatk and 7hati+3hatj-4hatk are coplanar.

If the position vectors of the vertices of a triangle are : vecA=hati+2hatj+3hatk, vecB=2hati+3hatj+hatk, vecC=3hati+hatj+2hatk , show that the triangle is an equilateral one.

The position vectors of A,B,C are 2hati+hatj-hatk,3hati-2hatj+hatk and hati+4hatj-3hatk respectively. Show that A, B and C are collinear.

Dot products of a vector with vertices hati+hatj-3hatk,hati+3hatj-2hatk and 2hati+hatj+4hatk are 0,5 and 8 respectively. Find the vector.

Show that the vectors hati-3hatj+2hatk,2hati-4hatj-hatk and 3hati+2hatj-hatk and linearly independent.