Question

\(\overline{a}\), \(\overline{b}\) are non-collinear vectors, \(|\overline{a}|=2\sqrt{2}\), \(|\overline{b}|=3\), and the angle between \(\overline{a}\) and \(\overline{b}\) is \(45^\circ\). Then the lengths of the diagonals of the parallelogram whose adjacent sides are represented by the vectors \(5\overline{a}+2\overline{b}\) and \(\overline{a}-3\overline{b}\) are

• A. \(15,593\)
• B. \(15,\sqrt{593}\)
• C. \(225,\sqrt{593}\)
• D. \(225,593\)

Hint:

For a parallelogram with adjacent side vectors \(\vec{u}\) and \(\vec{v}\), the diagonal vectors are \(\vec{u}+\vec{v}\) and \(\vec{u}-\vec{v}\).

Answer 1

The Correct Option is B

Step 1: Let the adjacent side vectors be defined.
Let \[ \vec{u}=5\overline{a}+2\overline{b} \] and \[ \vec{v}=\overline{a}-3\overline{b} \] The diagonals of the parallelogram are represented by \[ \vec{u}+\vec{v} \] and \[ \vec{u}-\vec{v}. \]

Step 2: Find the first diagonal vector.
\[ \vec{u}+\vec{v} = (5\overline{a}+2\overline{b})+(\overline{a}-3\overline{b}) \] \[ =6\overline{a}-\overline{b} \] So, the length of the first diagonal is \[ |6\overline{a}-\overline{b}|. \]

Step 3: Find the second diagonal vector.
\[ \vec{u}-\vec{v} = (5\overline{a}+2\overline{b})-(\overline{a}-3\overline{b}) \] \[ =4\overline{a}+5\overline{b} \] So, the length of the second diagonal is \[ |4\overline{a}+5\overline{b}|. \]

Step 4: Find \(\overline{a}\cdot\overline{b}\).
Given, \[ |\overline{a}|=2\sqrt{2},\qquad |\overline{b}|=3 \] and the angle between them is \[ 45^\circ. \] Therefore, \[ \overline{a}\cdot\overline{b} = |\overline{a}||\overline{b}|\cos45^\circ \] \[ =(2\sqrt{2})(3)\left(\frac{1}{\sqrt{2}}\right) \] \[ =6. \] Also, \[ |\overline{a}|^2=(2\sqrt{2})^2=8 \] and \[ |\overline{b}|^2=3^2=9. \]

Step 5: Find \(|6\overline{a}-\overline{b}|\).
\[ |6\overline{a}-\overline{b}|^2 = 36|\overline{a}|^2+|\overline{b}|^2-12(\overline{a}\cdot\overline{b}) \] \[ =36(8)+9-12(6) \] \[ =288+9-72 \] \[ =225. \] Therefore, \[ |6\overline{a}-\overline{b}|=15. \]

Step 6: Find \(|4\overline{a}+5\overline{b}|\).
\[ |4\overline{a}+5\overline{b}|^2 = 16|\overline{a}|^2+25|\overline{b}|^2+40(\overline{a}\cdot\overline{b}) \] \[ =16(8)+25(9)+40(6) \] \[ =128+225+240 \] \[ =593. \] Therefore, \[ |4\overline{a}+5\overline{b}|=\sqrt{593}. \]

Step 7: Final conclusion.
Thus, the lengths of the diagonals are \[ \boxed{15,\sqrt{593}} \]