Question

A parallelogram is constructed on $5 \bar { a } + 2 \bar { b }$ and $\bar { a } - 3 \bar { b }$ where |a| = 2$\sqrt { 2 }$ and |b| = 3. If the angle between is π/4, then the length of the longer diagonal is

• A. $\sqrt { 473 }$
• B. $\sqrt { 593 }$
• C. $\sqrt { 474 }$
• D. $\sqrt { 594 }$

Answer 1

Answer: (B)

$\sqrt { 593 }$

Answer 2

The vector representing one of the diagonals is

Hence the length of the diagonal = $\sqrt { ( 6 \bar { a } - \bar { b } ) \cdot ( 6 \bar { a } - \bar { b } ) }$

= $\sqrt { 36 | \overline { \mathrm { a } } | ^ { 2 } + | \overline { \mathrm { b } } | ^ { 2 } - 12 \overline { \mathrm { a } } \cdot \overline { \mathrm { b } } } = \sqrt { 36 \times 8 + 9 - 12 \times 3 \times 2 }$ =15. The other diagonal is $5 \bar { a } + 2 \bar { b } - \bar { a } + 3 \bar { b } = 4 \bar { a } + 5 \bar { b }$ .

Its length= $\sqrt { 16 | \overline { \mathrm { a } } | ^ { 2 } + 25 | \overline { \mathrm { b } } | ^ { 2 } + 40 \overline { \mathrm { a } } \cdot \overline { \mathrm { b } } }$

=$\sqrt { 128 + 225 + 40 \times 2 \times 3 }$= $\sqrt { 593 }$