Find the length diagonal AC of a prallelogram ABCD whose two adjacent sides AB and AD are represented respectively by` 2hati + 4hatj - 5hatk `and `hati + 2hatj + 3hatk `

Options

A. 5
B. 6
C. 7
D. 9

Answer

To find the length of diagonal AC of the parallelogram ABCD, we will follow these steps:

### Step 1: Identify the position vectors of the sides
We are given the position vectors of the adjacent sides:
- \( \vec{AB} = 2\hat{i} + 4\hat{j} - 5\hat{k} \)
- \( \vec{AD} = \hat{i} + 2\hat{j} + 3\hat{k} \)

### Step 2: Use the Law of Parallelogram
According to the law of addition for vectors in a parallelogram, the diagonal \( \vec{AC} \) can be found by adding the vectors \( \vec{AB} \) and \( \vec{AD} \):
\[ \vec{AC} = \vec{AB} + \vec{AD} \]

### Step 3: Perform the vector addition
Now, we will add the vectors:
\[ \vec{AC} = (2\hat{i} + 4\hat{j} - 5\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \]
Combining the components:
- For \( \hat{i} \): \( 2 + 1 = 3 \)
- For \( \hat{j} \): \( 4 + 2 = 6 \)
- For \( \hat{k} \): \( -5 + 3 = -2 \)

Thus, we have:
\[ \vec{AC} = 3\hat{i} + 6\hat{j} - 2\hat{k} \]

### Step 4: Find the length of diagonal AC
The length of the vector \( \vec{AC} \) can be calculated using the formula for the magnitude of a vector:
\[ |\vec{AC}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} \]
Calculating each term:
- \( (3)^2 = 9 \)
- \( (6)^2 = 36 \)
- \( (-2)^2 = 4 \)

Adding these values:
\[ |\vec{AC}| = \sqrt{9 + 36 + 4} = \sqrt{49} \]
Thus, we find:
\[ |\vec{AC}| = 7 \]

### Final Answer
The length of diagonal AC is \( 7 \) units.

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