Vectors along the adjacent sides of parallelogram are `veca = hati +2hatj +hatk and vecb = 2hati + 4hatj +hatk`. Find the length of the longer diagonal of the parallelogram.

Explanation

To find the length of the longer diagonal of the parallelogram formed by the vectors \(\vec{a} = \hat{i} + 2\hat{j} + \hat{k}\) and \(\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}\), we will follow these steps:

### Step 1: Identify the vectors
The vectors along the adjacent sides of the parallelogram are given as:
\[ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \]
\[ \vec{b} = 2\hat{i} + 4\hat{j} + \hat{k} \]

### Step 2: Find the diagonal vector
The diagonal \(\vec{AC}\) of the parallelogram can be found by adding the two vectors \(\vec{a}\) and \(\vec{b}\):
\[ \vec{AC} = \vec{a} + \vec{b} \]
Calculating this, we have:
\[ \vec{AC} = (\hat{i} + 2\hat{j} + \hat{k}) + (2\hat{i} + 4\hat{j} + \hat{k}) \]
\[ = (1 + 2)\hat{i} + (2 + 4)\hat{j} + (1 + 1)\hat{k} \]
\[ = 3\hat{i} + 6\hat{j} + 2\hat{k} \]

### Step 3: Calculate the magnitude of the diagonal vector
The length of the diagonal \(\vec{AC}\) is the magnitude of the vector \(\vec{AC}\):
\[ |\vec{AC}| = \sqrt{(3)^2 + (6)^2 + (2)^2} \]
Calculating this:
\[ |\vec{AC}| = \sqrt{9 + 36 + 4} \]
\[ = \sqrt{49} \]
\[ = 7 \]

### Conclusion
The length of the longer diagonal of the parallelogram is \(7\) units.

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