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

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.

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
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