A pyramid with vertex at point `P` has a regular hexagonal base `A B C D E F` , Position vector of points A and B are `hat i and hat i + 2 hat j` The centre of base has the position vector ` hat i+ hat j+sqrt(3) hat kdot` Altitude drawn from `P` on the base meets the diagonal `A D` at point `Gdot` find the all possible position vectors of `Gdot` It is given that the volume of the pyramid is `6sqrt(3)` cubic units and `A P` is 5 units.

Answer

`-(hati +2hatj+2sqrt3hatk), 3hati+2hatj +2sqrt3hatk`

Explanation

Let the centre of the base be (0). Therefore,
`" "|vec(AB)|=2`
`Delta OAB= (1)/(4)xx4xxsqrt3= sqrt3`
Base area `= 6sqrt3` sq. Units
Let height of the pyramid be h. Therefore,
`(1)/(3) xx 6sqrt3 h = 6 sqrt3 or h = 3units `

It is given that `|vec(AP)| = 5`. Therefore,
`" "AG= sqrt (25-9)= 4` units
`rArr |vec(AG)|= 4` units
Now `|vec(AG)| and |vec(AO)|` are collinear. Therefore
`" "vec(AG) = lamda vec(AO) rArr |vec(AG)| = |lamda||vec(AO)|`
or `" "2|lamda|= 4 or |lamda| = 2`
`rArr vec(AG)" "= pm 2(hati+hatj+ sqrt3hatk)`
`rArr vec(OG) = pm 2 (hati+hatj+ sqrt3hatk)+hati`
`" "= - (hatj+2hatj+ 2sqrt3hatk), 3hati+2hatj+2sqrt3hatk`