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`.

Answer 1

`|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`.