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Question
Two sides of a parallelogram are `3hat"i" + 4hat"j" - 5hat"k"` and `-2hat"j" + 7hat"k"`. Find the unit vectors parallel to the diagonals.
Solution
Let ABCD be a parallelogram with
`bar"AB" = 3hat"i" + 4hat"j" - 5hat"k" and bar"BC" = - 2hat"j" + 7hat"k"`
Then `bar"AC" = bar"AB" + bar"BC"`
`= (3hat"i" + 4hat"j" - 5hat"k") + (- 2hat"j" + 7hat"k")`
`= 3hat"i" + 2hat"j" + 2hat"k"`
∴ `|bar"AC"| = sqrt(3^2 + 2^2 + 2^2) =sqrt(9 + 4 + 4) = sqrt17`
∴ unit vector along `bar"AC" = bar"AC"/|bar"AC"|`
= `1/sqrt17 (3hat"i" + 2hat"j" + 2hat"k")`
Also, `bar"BD" = bar"BA" + bar"AD" = - bar"AB" + bar"BC" = bar"BC" - bar"AB"`
`= (- 2hat"j" + 7hat"k") - (3hat"i" + 4hat"j" - 5hat"k")`
`= - 3hat"i" - 6hat"j" + 12hat"k"`
`= 3(- hat"i" - 2hat"j" + 4hat"k")`
∴ `|bar"BD"| = 3sqrt((-1)^2 + (-2)^2 + 4^2) = 3sqrt(1 + 4 + 16) = 3sqrt21`
∴ unit vector along `bar"BD" = bar"BD"/|bar"BD"|`
`= (3(- hat"i" - 2hat"j" + 4hat"k"))/(3sqrt21)`
`= 1/sqrt21 (- hat"i" - 2hat"j" + 4hat"k")`
Hence, the unit vectors parallel to the diagonals are
`1/sqrt17 (3hat"i" + 2hat"j" + 2hat"k") and 1/sqrt21 (- hat"i" - 2hat"j" + 4hat"k")`
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