Question

Find the lengths of the sides of the triangle and also determine the type of a triangle:

L (3, -2, -3), M (7, 0, 1), N(1, 2, 1).

Answer 1

The position vectors bar"a", bar"b", bar"c" of the points L, M, N are

`bar"a" = 3hat"i" - 2hat"j" - 3hat"k",  bar"b" = 7hat"i" + hat"k",  bar"c" = hat"i" + 2hat"j" + hat"k"`

`bar"LM" = bar"b" - bar"a" = (7hat"i" + hat"k") - (3hat"i" - 2hat"j" - 3hat"k") = 4hat"i" + 2hat"j" + 4hat"k"`

`bar"MN" = bar"c" - bar"b" = (hat"i" + 2hat"j" + hat"k") - (7hat"i" + hat"k") = - 6hat"i" + 2hat"j"`

`bar"NL" = bar"a" - bar"c" = (3hat"i" - 2hat"j" - 3hat"k") - (hat"i" + 2hat"j" + hat"k")= 2hat"i" - 4hat"j" - 4hat"k"`

∴ l(LM) = `|bar"AB"| = sqrt(4^2 + 2^2 + 4^2) = sqrt(16 + 4 + 16) = sqrt36 = 6`

l(MN) = `|bar"MN"| = sqrt((- 6)^2 + 2^2) = sqrt(36 + 4) = sqrt40 = sqrt(10 xx 4) = 2sqrt10`

l(NL) = `|bar"NL"| = sqrt((2)^2 + (-4)^2 + (- 4)^2) = sqrt(4 + 16 + 16) = sqrt36 = 6` 

l(LM) = 6, l(MN) = `2sqrt10`, l(NL) = 6

∴ Δ LMN is isosceles.