Advertisement Remove all ads

If a parallelogram is constructed on the vectors apqbpqandpqa¯=3p¯-q¯,b¯=p¯+3q¯and|p¯|=|q¯|=2 and angle between pandqp¯andq¯ is π3, and angle between lengths of the sides is 7:13. - Mathematics and Statistics

Advertisement Remove all ads
Advertisement Remove all ads
Sum

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`.

Advertisement Remove all ads

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`.

Concept: Vectors and Their Types
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×