If the vectors `(hat(i) + hat(j) + hat(k))` and `3hat(i)` form two sides of a triangle, then area of the triangle is :

Options

A. `sqrt(3)` unit
B. `2sqrt(3)` unit
C. `(3)/(sqrt(2))` unit
D. `3sqrt(2)` unit

Answer

To find the area of the triangle formed by the vectors \( \vec{A} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{B} = 3\hat{i} \), we can use the formula for the area of a triangle given by two vectors:

\[ \text{Area} = \frac{1}{2} |\vec{A} \times \vec{B}| \]

### Step 1: Write down the vectors
We have:
- \( \vec{A} = \hat{i} + \hat{j} + \hat{k} \)
- \( \vec{B} = 3\hat{i} \)

### Step 2: Set up the cross product
The cross product \( \vec{A} \times \vec{B} \) can be calculated using the determinant of a matrix formed by the unit vectors and the components of the vectors:

\[ \vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ 3 & 0 & 0 \end{vmatrix} \]

### Step 3: Calculate the determinant
Using the determinant, we can calculate:

\[ \vec{A} \times \vec{B} = \hat{i} \begin{vmatrix} 1 & 1 \\ 0 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 1 \\ 3 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 3 & 0 \end{vmatrix} \]

Calculating the determinants:

1. For \( \hat{i} \):
\[ \begin{vmatrix} 1 & 1 \\ 0 & 0 \end{vmatrix} = 0 \]

2. For \( \hat{j} \):
\[ \begin{vmatrix} 1 & 1 \\ 3 & 0 \end{vmatrix} = (1)(0) - (1)(3) = -3 \]

3. For \( \hat{k} \):
\[ \begin{vmatrix} 1 & 1 \\ 3 & 0 \end{vmatrix} = (1)(0) - (1)(3) = -3 \]

Putting it all together:
\[ \vec{A} \times \vec{B} = 0\hat{i} + 3\hat{j} - 3\hat{k} = 3\hat{j} - 3\hat{k} \]

### Step 4: Find the magnitude of the cross product
Now we find the magnitude of \( \vec{A} \times \vec{B} \):
\[ |\vec{A} \times \vec{B}| = \sqrt{(0)^2 + (3)^2 + (-3)^2} = \sqrt{0 + 9 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 5: Calculate the area of the triangle
Now we can calculate the area:
\[ \text{Area} = \frac{1}{2} |\vec{A} \times \vec{B}| = \frac{1}{2} (3\sqrt{2}) = \frac{3\sqrt{2}}{2} \]

Thus, the area of the triangle formed by the vectors is:
\[ \text{Area} = \frac{3\sqrt{2}}{2} \]