The adjacent sides AB and AC of a triangle ABC are represented by the vectors `-2hat(i)+ 3hat(j) + 2hat(k) and -4hat(i) + 5hat(j) + 2hat(k)` respectively. The area of the triangle ABC is

Options

A. 6 sq units
B. 5 sq units
C. 4 sq units
D. 3 sq units

Answer

To find the area of triangle ABC given the vectors representing the adjacent sides AB and AC, we can follow these steps:

### Step 1: Identify the vectors
The vectors representing the sides of the triangle are:
- AB = \(-2\hat{i} + 3\hat{j} + 2\hat{k}\)
- AC = \(-4\hat{i} + 5\hat{j} + 2\hat{k}\)

### Step 2: Use the formula for the area of a triangle
The area \(A\) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} | \mathbf{AB} \times \mathbf{AC} | \]
where \(\times\) denotes the cross product.

### Step 3: Calculate the cross product \(\mathbf{AB} \times \mathbf{AC}\)
To find the cross product, we can use the determinant of a matrix formed by the unit vectors \(\hat{i}, \hat{j}, \hat{k}\) and the components of the vectors AB and AC:
\[ \mathbf{AB} \times \mathbf{AC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -2 & 3 & 2 \\ -4 & 5 & 2 \end{vmatrix} \]

### Step 4: Compute the determinant
Calculating the determinant:
\[ \mathbf{AB} \times \mathbf{AC} = \hat{i} \begin{vmatrix} 3 & 2 \\ 5 & 2 \end{vmatrix} - \hat{j} \begin{vmatrix} -2 & 2 \\ -4 & 2 \end{vmatrix} + \hat{k} \begin{vmatrix} -2 & 3 \\ -4 & 5 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. \(\begin{vmatrix} 3 & 2 \\ 5 & 2 \end{vmatrix} = (3)(2) - (5)(2) = 6 - 10 = -4\)
2. \(\begin{vmatrix} -2 & 2 \\ -4 & 2 \end{vmatrix} = (-2)(2) - (-4)(2) = -4 + 8 = 4\)
3. \(\begin{vmatrix} -2 & 3 \\ -4 & 5 \end{vmatrix} = (-2)(5) - (-4)(3) = -10 + 12 = 2\)

Putting it all together:
\[ \mathbf{AB} \times \mathbf{AC} = -4\hat{i} - 4\hat{j} + 2\hat{k} \]

### Step 5: Calculate the magnitude of the cross product
Now, we find the magnitude of the vector:
\[ |\mathbf{AB} \times \mathbf{AC}| = \sqrt{(-4)^2 + (-4)^2 + (2)^2} = \sqrt{16 + 16 + 4} = \sqrt{36} = 6 \]

### Step 6: Calculate the area of the triangle
Finally, substituting back into the area formula:
\[ A = \frac{1}{2} \times 6 = 3 \]

### Final Answer
The area of triangle ABC is \(3\) square units.

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