Question

Two sides of a triangle OAB are given by:
\(\overrightarrow{𝑂𝐴} = 𝑥̂ + 2𝑦̂ + 𝑧̂ \)
\(\overrightarrow{𝑂B}= 2𝑥̂ − 𝑦̂ + 3𝑧̂ \)
The area of the triangle is _______. (Rounded off to one decimal place)

Answer 1

Correct Answer: 4.2

Area of the Triangle OAB 

The area of the triangle OAB is given by:

Area = (1/2) |OA × OB|

where OA × OB is the cross product of the vectors OA and OB.

Step 1: Compute the Cross Product

Given vectors:

\[ \overrightarrow{OA} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad \overrightarrow{OB} = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} \]

The cross product is:

\[ \overrightarrow{OA} \times \overrightarrow{OB} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 1 \\ 2 & -1 & 3 \end{vmatrix} \]

Step 2: Expand the Determinant

\[ \overrightarrow{OA} \times \overrightarrow{OB} = \hat{i} \begin{vmatrix} 2 & 1 \\ -1 & 3 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 1 \\ 2 & 3 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 2 & -1 \end{vmatrix} \]

Computing each determinant:

\[ \hat{i} ((2)(3) - (1)(-1)) - \hat{j} ((1)(3) - (1)(2)) + \hat{k} ((1)(-1) - (2)(2)) \]

\[ = \hat{i} (6 + 1) - \hat{j} (3 - 2) + \hat{k} (-1 - 4) \]

\[ = 7\hat{i} - \hat{j} - 5\hat{k} \]

Step 3: Compute the Magnitude

The magnitude of the cross product is:

\[ |\overrightarrow{OA} \times \overrightarrow{OB}| = \sqrt{(7)^2 + (-1)^2 + (-5)^2} \]

\[ = \sqrt{49 + 1 + 25} = \sqrt{75} = 8.66 \]

Step 4: Compute the Area

\[ \text{Area} = \frac{1}{2} \times 8.66 = 4.33 \]

Final Answer

The area of the triangle OAB is 4.33.