# Two sides of a parallelogram are ijk3i^+4j^-5k^ and jk-2j^+7k^. Find unit vectors parallel to the diagonals. - Mathematics and Statistics

Sum

Two sides of a parallelogram are 3hat"i" + 4hat"j" - 5hat"k" and  -2hat"j" + 7hat"k". Find 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"i" + 7hat"k"

Then bar"AC" = bar"AB" + bar"BC"

= (3hat"i" + 4hat"j" - 5hat"k") + (- 2hat"i" + 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"i" + 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")

Concept: Vectors and Their Types
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