Vectors along the adjacent sides of parallelogram are `veca = 2hati +4hatj -5hatk and vecb = hati + 2hatj +3hatk`. 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 **a** and **b**, we can follow these steps:

### Step 1: Identify the given vectors
The vectors along the adjacent sides of the parallelogram are:
\[ \vec{a} = 2\hat{i} + 4\hat{j} - 5\hat{k} \]
\[ \vec{b} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 2: Find the diagonal vector
The longer diagonal of the parallelogram can be represented by the vector from point A to point C, denoted as \(\vec{AC}\). This vector can be found by adding the vectors \(\vec{a}\) and \(\vec{b}\):
\[ \vec{AC} = \vec{a} + \vec{b} \]

### Step 3: Perform the vector addition
Now, we will add the components of 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: Calculate the magnitude of the diagonal vector
The length of the diagonal \(\vec{AC}\) can be found using the magnitude formula:
\[ |\vec{AC}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} \]
Calculating each term:
- \(3^2 = 9\)
- \(6^2 = 36\)
- \((-2)^2 = 4\)

Now, sum these values:
\[ |\vec{AC}| = \sqrt{9 + 36 + 4} = \sqrt{49} \]

### Step 5: Final calculation
Taking the square root gives us:
\[ |\vec{AC}| = 7 \]

### Conclusion
The length of the longer diagonal of the parallelogram is **7 units**.

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