Question

If line joining points A and B having position vectors `6overlinea - 4overlineb + 4overlinec` and `-4overlinec` respectively, and the line joining the points C and D having position vectors `-overlinea - 2overlineb - 3overlinec` and `overlinea + 2overlineb - 5overlinec` intersect, then their point of intersection is ______

पर्याय

• B

• C

• D

• A

Answer 1

If line joining points A and B having position vectors `6overlinea - 4overlineb + 4overlinec` and `-4overlinec` respectively, and the line joining the points C and D having position vectors `-overlinea - 2overlineb - 3overlinec` and `overlinea + 2overlineb - 5overlinec` intersect, then their point of intersection is B.

Explanation:

Equation of line AB in vector form is

`overliner = 6overlinea - 4overlineb + 4overlinec + lambda(-4overlinec - {6overlinea - 4overlineb + 4overlinec})`

⇒ `overliner = 6overlinea - 4overlineb + 4overlinec + lambda(-6overlinea + 4overlineb - 8overlinec)` .............(i)

Equation of line CD in vector form is

`overline{r^'} = overlinea + 2overlineb - 5overlinec + lambda^'(-overlinea - 2overlineb - 3overlinec - {overlinea + 2overlineb - 5overlinec})`

⇒ `overline{r^'} = overlinea + 2overlineb - 5overlinec + lambda^'(-2overlinea - 4overlineb + 2overlinec)` .....(ii)

The point of intersection of AB and CD will satisfy

`overliner = overline{r^'}`

⇒ `6overlinea - 4overlineb + 4overlinec + lambda(-6overlinea + 4overlineb - 8overlinec) = overlinea + 2overlineb - 5overlinec + lambda^'(-2overlinea - 4overlineb + 2overlinec)`

Comparing the coefficients of `overlinea` and `overlineb`, we get

6λ - 2λ' = 5 ....................(iii)
2λ + 2λ' = 3 ....................(iv)

⇒ λ = 1 and λ' = `1/2`

Substituting the value of λ in equation (i), we get the point of intersection

∴ Point of intersection `overliner = -4overlinec` i.e. point B.