Question

Show that the vectors `- 2hat"i" - hat"j" - hat"k", - 3hat"i" - 4hat"j" - 4hat"k", hat"i" - 3hat"j" - 5hat"k"` form a right angled triangle

Answer 1

Let the given vectors be `vec"AB" = 2hat"i" - hat"j" + hat"k"`

`vec"BC" = 3hat"i" - 4hat"j" - 4hat"k"` and `vec"AC" = hat"i" - 3hat"j" - hat"k"`

`|vec"AB"| = |2hat"i" - hat"j" + hat"k"|`

AB = `sqrt(2^2 + (-1)^2 + 1^2)`

= `sqrt(4 + 1 + 1)`

AB = `sqrt(6)`

`|vec"BC"| = |3hat"i" - 4hat"j" - 4hat"k"|`

BC = `sqrt(3^2 + (-4)^2 + (-4)^2`

= `sqrt(9 + 16 + 16)`

BC = `sqrt(41)`

`|vec"AC"| = |hat"i" - 3hat"j" - 5hat"k"|`

AC = `sqrt(1^2 + (-3)^2 + (-5)^2`

= `sqrt(1 +9 + 25)`

AC = `sqrt(35)`

AB² + AC² = 6 + 35 = 41 .......(1)

BC² = 41   .......(2)

From equation (1) and (2), we get

AB² + AC² = BC²  

∴ The given vectors from  right anled triange.