let `|a|=2sqrt(2),|b|=3` and the angle between a and b is `(pi)/(4)`. If a parallelogram is constructed with adjacent sides `2a-3b and a+b`, then its longer diagonal is of length

Options

A. 10
B. 8
C. `2sqrt(26)`
D. 6

Answer

To find the length of the longer diagonal of the parallelogram constructed with adjacent sides \( \mathbf{R_1} = 2\mathbf{a} - 3\mathbf{b} \) and \( \mathbf{R_2} = \mathbf{a} + \mathbf{b} \), we will follow these steps:

### Step 1: Calculate the Magnitude of \( \mathbf{R_1} \)

The magnitude of \( \mathbf{R_1} \) can be calculated using the formula:
\[ |\mathbf{R_1}| = |2\mathbf{a} - 3\mathbf{b}| \]
Using the formula for the magnitude of a vector:
\[ |\mathbf{X} - \mathbf{Y}| = \sqrt{|\mathbf{X}|^2 + |\mathbf{Y}|^2 - 2(\mathbf{X} \cdot \mathbf{Y})} \]
we can write:
\[ |\mathbf{R_1}| = \sqrt{|2\mathbf{a}|^2 + |-3\mathbf{b}|^2 - 2(2\mathbf{a} \cdot (-3\mathbf{b}))} \]

Calculating each term:
- \( |2\mathbf{a}| = 2|\mathbf{a}| = 2 \times 2\sqrt{2} = 4\sqrt{2} \)
- \( |-3\mathbf{b}| = 3|\mathbf{b}| = 3 \times 3 = 9 \)

Now substituting these values:
\[ |\mathbf{R_1}| = \sqrt{(4\sqrt{2})^2 + 9^2 + 12(\mathbf{a} \cdot \mathbf{b})} \]
Next, we need to calculate \( \mathbf{a} \cdot \mathbf{b} \):
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta = (2\sqrt{2})(3)\cos\left(\frac{\pi}{4}\right) = 6\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 6 \]

Now substituting \( \mathbf{a} \cdot \mathbf{b} \):
\[ |\mathbf{R_1}| = \sqrt{32 + 81 + 12 \times 6} = \sqrt{32 + 81 + 72} = \sqrt{185} \]

### Step 2: Calculate the Magnitude of \( \mathbf{R_2} \)

Now we calculate the magnitude of \( \mathbf{R_2} \):
\[ |\mathbf{R_2}| = |\mathbf{a} + \mathbf{b}| \]
Using the formula for the magnitude of a vector:
\[ |\mathbf{R_2}| = \sqrt{|\mathbf{a}|^2 + |\mathbf{b}|^2 + 2(\mathbf{a} \cdot \mathbf{b})} \]
Substituting the known values:
\[ |\mathbf{R_2}| = \sqrt{(2\sqrt{2})^2 + 3^2 + 2 \times 6} = \sqrt{8 + 9 + 12} = \sqrt{29} \]

### Step 3: Calculate the Dot Product \( \mathbf{R_1} \cdot \mathbf{R_2} \)

Next, we calculate the dot product:
\[ \mathbf{R_1} \cdot \mathbf{R_2} = (2\mathbf{a} - 3\mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) \]
Expanding this:
\[ = 2\mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} - 3\mathbf{b} \cdot \mathbf{a} - 3\mathbf{b} \cdot \mathbf{b} \]
Substituting the values:
\[ = 2|\mathbf{a}|^2 + 2(\mathbf{a} \cdot \mathbf{b}) - 3(\mathbf{a} \cdot \mathbf{b}) - 3|\mathbf{b}|^2 \]
\[ = 2(8) + 2(6) - 3(6) - 3(9) = 16 + 12 - 18 - 27 = -17 \]

### Step 4: Calculate the Length of the Longer Diagonal

The lengths of the diagonals of the parallelogram can be calculated using the formula:
\[ |\mathbf{D_1}| = |\mathbf{R_1} + \mathbf{R_2}| \]
\[ |\mathbf{D_2}| = |\mathbf{R_2} - \mathbf{R_1}| \]

For the longer diagonal, we will calculate \( |\mathbf{D_2}| \):
\[ |\mathbf{D_2}| = \sqrt{|\mathbf{R_2}|^2 + |\mathbf{R_1}|^2 - 2(\mathbf{R_1} \cdot \mathbf{R_2})} \]
Substituting the values:
\[ = \sqrt{29 + 185 + 34} = \sqrt{248} = \sqrt{4 \times 62} = 2\sqrt{62} \]

### Final Answer

The length of the longer diagonal is:
\[ \boxed{2\sqrt{62}} \]