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Question
If a parallelogram is constructed on the vectors `bar"a" = 3bar"p" - bar"q", bar"b" = bar"p" + 3bar"q" and |bar"p"| = |bar"q"| = 2` and angle between `bar"p" and bar"q"` is `pi/3,` and angle between lengths of the sides is `sqrt7 : sqrt13`.
Solution
`|bar"p"| = |bar"q"| = 2` and angle between `bar"p" and bar"q"` is `pi/3`.
∴ `bar"p".bar"q" = |bar"p"||bar"q"| "cos" pi/3 = 2xx2xx1/2 = 2`
Now, `bar"a" = 3bar"p" - bar"q"`
∴ `|bar"a"|^2 = |(3bar"p" - bar"q")|^2`
`= (3bar"p" - bar"q").(3bar"p" - bar"q")`
`= 3bar"p".(3bar"p" - bar"q") - bar"q".(3bar"p" - bar"q")`
`= 9bar"p".bar"p" - 3bar"p".bar"q" - 3bar"q".bar"p" + bar"q".bar"q"`
`= 9|bar"p"|^2 - 6bar"p".bar"q" + |bar"q"|^2` .....`[∵ bar"q".bar"p" = bar"p".bar"q"]`
`= 9xx4 - 6xx2 + 4 .......[∵ bar"p"bar"q" = 2]`
= 28
∴ `|bar"a"| = sqrt28`
Also `bar"b" = bar"p" + 3bar"q"`
∴ `|bar"b"|^2 = |bar"p" + 3bar"q"|^2`
`= (bar"p" + 3bar"q").(bar"p" + 3bar"q")`
`= bar"p"(bar"p" + 3bar"q") + 3bar"q"(bar"p" + 3bar"q")`
`= bar"p".bar"p" + 3bar"p".bar"q" - 3bar"q".bar"p" + 9bar"q".bar"q" ......[∵ bar"p".bar"q" = bar"q".bar"p"]`
`= |bar"p"|^2 + 3bar"p""q" + 3bar"p".bar"q" + 9 |bar"q"|^2`
= 4 + 12 + 36 ......`[∵ bar"p".bar"q" = 2]`
= 52
∴ `|bar"b"| = sqrt52`
Ratio of lengths of the sides
`= |bar"a"|/|bar"b"| = sqrt28/sqrt52 = (2sqrt7)/(2sqrt13) = sqrt7/sqrt13`.
Hence, the ratio of the lengths of the sides is `sqrt7 : sqrt13`.
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