Question

Find the area of the triangle whose sides are given by vectors: \[ \vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}, \quad \vec{b} = \hat{i} - \hat{j} + 3\hat{k} \]

Hint:

The area of a triangle formed by two vectors can be calculated using the formula \( \frac{1}{2} |\vec{a} \times \vec{b}| \).

Answer 1

Step 1: Formula for the area of the triangle.
The area of a triangle whose sides are given by vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ \text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}| \] Step 2: Finding the cross product.
The cross product \( \vec{a} \times \vec{b} \) is calculated as follows: \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
2 & 3 & -1
1 & -1 & 3 \end{vmatrix} \] Expanding the determinant: \[ \vec{a} \times \vec{b} = \hat{i} \begin{vmatrix} 3 & -1
-1 & 3 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & -1
1 & 3 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 3
1 & -1 \end{vmatrix} \] \[ = \hat{i} (9 + 1) - \hat{j} (6 + 1) + \hat{k} (-2 - 3) \] \[ = 10\hat{i} - 7\hat{j} - 5\hat{k} \] Step 3: Finding the magnitude of the cross product.
The magnitude of \( \vec{a} \times \vec{b} \) is: \[ |\vec{a} \times \vec{b}| = \sqrt{10^2 + (-7)^2 + (-5)^2} = \sqrt{100 + 49 + 25} = \sqrt{174} \] Step 4: Calculating the area.
Thus, the area of the triangle is: \[ \text{Area} = \frac{1}{2} \sqrt{174} \] Step 5: Conclusion.
Therefore, the area of the triangle is \( \frac{1}{2} \sqrt{174} \).