Find the area of the triangle whose adjacent sides are determined by the vectors `vec(a)=(-2 hat(i)-5 hat(k)) and vec(b)= ( hat(i)- 2 hat(j) - hat(k))`.

Answer

To find the area of the triangle whose adjacent sides are determined by the vectors \(\vec{a} = -2\hat{i} - 5\hat{k}\) and \(\vec{b} = \hat{i} - 2\hat{j} - \hat{k}\), we can follow these steps:

### Step 1: Write down the vectors
We have:
\[ \vec{a} = -2\hat{i} + 0\hat{j} - 5\hat{k} \]
\[ \vec{b} = 1\hat{i} - 2\hat{j} - 1\hat{k} \]

### Step 2: Find the cross product \(\vec{a} \times \vec{b}\)
The area of the triangle is given by:
\[ \text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}| \]
To find \(\vec{a} \times \vec{b}\), we can use the determinant of a matrix formed by the unit vectors and the components of \(\vec{a}\) and \(\vec{b}\):
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -2 & 0 & -5 \\ 1 & -2 & -1 \end{vmatrix} \]

### Step 3: Calculate the determinant
Using the determinant formula:
\[ \vec{a} \times \vec{b} = \hat{i} \begin{vmatrix} 0 & -5 \\ -2 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} -2 & -5 \\ 1 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} -2 & 0 \\ 1 & -2 \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \(\begin{vmatrix} 0 & -5 \\ -2 & -1 \end{vmatrix} = (0)(-1) - (-5)(-2) = 0 - 10 = -10\)
2. \(\begin{vmatrix} -2 & -5 \\ 1 & -1 \end{vmatrix} = (-2)(-1) - (-5)(1) = 2 + 5 = 7\)
3. \(\begin{vmatrix} -2 & 0 \\ 1 & -2 \end{vmatrix} = (-2)(-2) - (0)(1) = 4 - 0 = 4\)

Now substituting back:
\[ \vec{a} \times \vec{b} = -10\hat{i} - 7\hat{j} + 4\hat{k} \]

### Step 4: Find the magnitude of \(\vec{a} \times \vec{b}\)
Now, we find the magnitude:
\[ |\vec{a} \times \vec{b}| = \sqrt{(-10)^2 + (-7)^2 + (4)^2} = \sqrt{100 + 49 + 16} = \sqrt{165} \]

### Step 5: Calculate the area of the triangle
Finally, the area of the triangle is:
\[ \text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}| = \frac{1}{2} \sqrt{165} \]

### Final Answer
The area of the triangle is:
\[ \text{Area} = \frac{\sqrt{165}}{2} \text{ square units} \]

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