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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
Solution
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.
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