Home
Class 11
MATHS
Consider a triangular pyramid ABCD the p...

Consider a triangular pyramid ABCD the position vectors of whone agular points are `A(3,0,1),B(-1,4,1),C(5,3, 2) and D(0,-5,4)` Let G be the point of intersection of the medians of the triangle BCT. The length of the vector `bar(AG)` is

A

`sqrt17`

B

`sqrt51//3`

C

`3//sqrt6`

D

`sqrt59//4`

Text Solution

Verified by Experts

The correct Answer is:
b
Point G is `( 4/3, 1/3,8/3) `. Therefore,
`|vec(AG)|^(2)= (5/3)^(2)+ 1/9+(5/2)^(2)= 51/9`
`or |vec(AG)|= sqrt51/3`
`vec(AB)= -4hati + 4hatj + 0hatk`
` vec(AC)= 2hati + 2hatj + 2hatk`
`vec(AB)xxvec(AC)= -8-8|{:(hati,hatj,hatk),(1,-1,0),(1,1,1):}|`
`8 (hati + hatj - 2hatk)`
Area of `triangleABC = 1/2|vec(AB)xx vec(AC)|=4 sqrt6`
`vec(AD)= -3hati - 5hatj + 3hatk`
The length of the perpendicular from the vertex D on the opposite face
`=|" projection of " vec(AD) " on "vec(AB)xx vec(AC)|`
`|((-3hati-5hatj+ 3hatk)(hati+hatj-2hatk))/sqrt6|`
`|(-3-5-6)/sqrt6|=14/sqrt6`
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

Find the point of intersection of AB and A( 6,-7,0),B(16,-19,-4,) , C(0,3,-6) and D(2,-5,10).

The vertices of the rectangle ABCD are A(-1,0), B(2, 0), C(a, b) and D(-1, 4). Then the length of the diagonal AC is

Find the position vector of the midpoint of the vector joining the points P(2, 3, 4) and Q(4, 1 -2)

The point A(1,2), B(2,-3), C(-1,-5) and D(-2, 4) in order are the vertices of

Find the position vector of the mid point of the vector joining the points P(2,3,4) and Q(4,1,-2).

The volume of a right triangular prism ABCA_1B_1C_1 is equal to 3. If the position vectors of the vertices of thebase ABC are A(1, 0, 1), B(2,0, 0) and C(O, 1, 0) , then position vectors of the vertex A_1 , can be

A triangle with vertices (4, 0), (-1, -1), (3, 5) is

Consider the points A(2,2,-1) , B(3,4,2) and C(7,0,6) i.Find AB. ii. Find the vector and cartesian equation of the plane passing through these points .

Find the area of the triangle formed by the points A(0, – 1), B(– 2, 6) and C(– 3, – 5)