Angle between diagonals of a parallelogram whose side are represented by `veca=2hati+hatj+hatk` and `vecb=hati-hatj-hatk`

Options

A. `cos^(-1)((1)/(3))`
B. `cos^(-1)((1)/(2))`
C. `cos^(-1)((4)/(9))`
D. `cos^(-1)((5)/(9))`

Answer

To find the angle between the diagonals of a parallelogram whose sides are represented by the vectors \(\vec{a} = 2\hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} - \hat{j} - \hat{k}\), we can follow these steps:

### Step 1: Find the diagonals of the parallelogram
The diagonals of a parallelogram can be represented as:
- First diagonal, \(\vec{D_1} = \vec{a} + \vec{b}\)
- Second diagonal, \(\vec{D_2} = \vec{a} - \vec{b}\)

### Step 2: Calculate \(\vec{D_1}\) and \(\vec{D_2}\)
1. **Calculate \(\vec{D_1}\)**:
\[ \vec{D_1} = (2\hat{i} + \hat{j} + \hat{k}) + (\hat{i} - \hat{j} - \hat{k}) \]
\[ = (2 + 1)\hat{i} + (1 - 1)\hat{j} + (1 - 1)\hat{k} \]
\[ = 3\hat{i} + 0\hat{j} + 0\hat{k} = 3\hat{i} \]

2. **Calculate \(\vec{D_2}\)**:
\[ \vec{D_2} = (2\hat{i} + \hat{j} + \hat{k}) - (\hat{i} - \hat{j} - \hat{k}) \]
\[ = (2 - 1)\hat{i} + (1 + 1)\hat{j} + (1 + 1)\hat{k} \]
\[ = 1\hat{i} + 2\hat{j} + 2\hat{k} \]

### Step 3: Find the angle between the diagonals
To find the angle \(\theta\) between the two vectors \(\vec{D_1}\) and \(\vec{D_2}\), we use the formula:
\[ \cos \theta = \frac{\vec{D_1} \cdot \vec{D_2}}{|\vec{D_1}| |\vec{D_2}|} \]

### Step 4: Calculate the dot product \(\vec{D_1} \cdot \vec{D_2}\)
\[ \vec{D_1} \cdot \vec{D_2} = (3\hat{i}) \cdot (1\hat{i} + 2\hat{j} + 2\hat{k}) \]
\[ = 3 \cdot 1 + 0 + 0 = 3 \]

### Step 5: Calculate the magnitudes \(|\vec{D_1}|\) and \(|\vec{D_2}|\)
1. **Magnitude of \(\vec{D_1}\)**:
\[ |\vec{D_1}| = |3\hat{i}| = 3 \]

2. **Magnitude of \(\vec{D_2}\)**:
\[ |\vec{D_2}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 6: Substitute values into the cosine formula
\[ \cos \theta = \frac{3}{3 \cdot 3} = \frac{3}{9} = \frac{1}{3} \]

### Step 7: Find the angle \(\theta\)
\[ \theta = \cos^{-1}\left(\frac{1}{3}\right) \]

### Final Answer
The angle between the diagonals of the parallelogram is \(\cos^{-1}\left(\frac{1}{3}\right)\).

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