Home
Class 11
MATHS
The position vectors of the vertices A, ...

The position vectors of the vertices A, B and C of a tetrahedron ABCD are `hat i + hat j + hat k`, `hat k `, `hat i` and `hat 3i`,respectively. The altitude from vertex D to the opposite face ABC meets the median line through Aof triangle ABC at a point E. If the length of the side AD is 4 and the volume of the tetrahedron is2/2/3, find the position vectors of the point E for all its possible positfons

Text Solution

Verified by Experts

The correct Answer is:
`-hati +_3hatj + 3hatk or 3hati-hatj-hatk`
we are given AD=4
Volume of tetrahedron = `(2sqrt2)/3`
`Rightarrow 1/3 (" Area of "triangleABC)p=(2sqrt2)/3`
`1/2|vec(BA) xx vec(BC)|p=2sqrt2`
`1/2 |(hatj +hatk) xx2hati|p=2sqrt2`
`or |hatj - hatk| p= 2 sqrt2`
`or sqrt2 p =2 sqrt2or p=2`
we have to find the P.V of point E.Let it divide median AF in the ratio `lambda:1`
` P.V. "of" E = (lambda .2hati + (hati +hatj +hatk)/(lambda +1)`
`vec(AE)` P.V or E - P.V. of `A=(lambda(hati-hatj-hatk))/(lamda+1)`
`|vec(AE)|^(2)=3(lamda/(lambda+1))^(2)`
Now ` 4+3 (lamda/(lambda + 1))=16`
`(lamda/(lambda+_1))=+-2`
` lamda=-2 or -2//3`
Putting the value of `lamda` in (i), we get the P.V. of possible
Positions of E as `-hati+3hatj+3hatk or 3hati-hatj-hatk`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise 2.1|18 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise 2.2|15 Videos

  • CONIC SECTIONS

    CENGAGE|Exercise Solved Examples And Exercises|1255 Videos

  • LIMITS AND DERIVATIVES

    CENGAGE|Exercise Solved Examples And Exercises|288 Videos

Similar Questions

Explore conceptually related problems

The position vectors of the vertices A ,B ,a n dC of a triangle are hat i+ hat j , hat j+ hat ka n d hat i+ hat k , respectively. Find the unite vector hat r lying in the plane of A B C and perpendicular to I A ,w h e r eI is the incentre of the triangle.

Find the angle between the vectors hat i - 2 hat j + 3 hat k and 3 hat i - 2 hat j + hat k .

If two side of a triangle are hat i+2 hat ja n d hat i+ hat k , then find the length of the third side.

The unit vector in the direction of the sum of vectors hat i + hat j + hat k and 2 hat i + 3 hat j + 4 hat k is

The position vectors of point A ,B ,a n dC are hat i+ hat j+ hat k , hat i+5 hat j- hat ka n d2 hat i+3 hat j+5 hat k , respectively. Then greatest angel of triangle A B C is 120^0 b. 90^0 c. cos^(-1)(3//4) d. none of these

Show that the points with position vectors 2 hat i + 6 hat j + 3 hat k , hat i + 2 hat j + 7 hat k and 3 hat i + 10 hat j - hat k are collinear.

The position vectors of the point A ,B ,Ca n dDa r e3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k ,- hat i+ hat j+2 hat k and 4 hat i+5 hat j+lambda hat k , respectively. If the points A ,B ,Ca n dD lie on a plane, find the value of lambdadot

Prove that the vectors 3 hat i + hat j + 3 hat k and hat i - hat k are perpendicular.

Find the projection of the vector hat i + 3 hat j + 7 hat k on the vector 7 hat i - hat j + 8 hat k .

The position vectors of the vertices of DeltaABC are 3 hat i - 4 hat j - 4 hat k , 2 hat i - hat j + hat k and hat i - 3 hat j - 5 hat k respectively. (i) Find vec (AB), vec(BC) and vec(CA) . (ii) Prove that DeltaABC is a right angled triangle. (iii) Find the unit vector perpendicular to both vec (AB) and vec(BC) .