Question

Prove by vector method that the diagonals of a rhombus bisect each other at right angles

Answer 1

Let ABCD be a rhombus

To prove `bar"AC" * bar"BD"` = 0

We have AB = BC = CD = DA

Now `bar"AC" = bar"AB" + "BC"`

`bar"BD" = bar"BC" + bar"CD"`

= `bar"BC" - bar"AB"`  .......`("Since"  bar"CD" = - bar"AB")`

`bar"AC" * bar"BD" = (bar"BC" + bar"AB") * (bar"BC" - bar"AB")`

= `(bar"BC")^2 - (bar"AB")^2`

= `("BC")^2 - ("AB")^2`

`bar"AC" * bar"BD"` = 0

∴ `bar"AC"` ⊥' to `bar"BD"`

Hence the result.