Question

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

A(2, -1, 0), B(4, 1, 1), C(4, -5, 4)

Answer 1

The position vectors bar"a", bar"b", bar"c" of the points A, B, C are

`bar"a" = 2hat"i" - hat"j"  ,bar"b" = 4hat"i" + hat"j" + hat"k", bar"c" = 4hat"i" - 5hat"j" + 4hat"k"`

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

`bar"BC" = bar"c" - bar"b" = (4hat"i" - 5hat"j" + 4hat"k") - (4hat"i" + hat"j" + hat"k") = - 6hat"j" + 3hat"k"`

`bar"CA" = bar"a" - bar"c" = (2hat"i" - hat"j") - (4hat"i" - 5hat"j" + 4hat"k") = - 2hat"i" + 4hat"j" - 4hat"k"`

∴ l(AB) = `|bar"AB"| = sqrt(2^2 + 2^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt9 = 3`

l(BC) = `|bar"BC"| = sqrt((- 6)^2 + 3^2) = sqrt(36 + 9) = sqrt45 = 3sqrt5`

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

∴ [l(AB)]² + [l(CA)]² = 3² + 6² = 9 + 36 = 45 = `(3sqrt5)^2`

= [l(BC)]² 

∴ Δ ABC is right-angled at A.