Vector Algebra

Step 1: Formula for the area of the triangle.
The area of a triangle whose sides are given by vectors and is given by: Step 2: Finding the cross product.
The cross product is calculated as follows: Expanding the determinant: Step 3: Finding the magnitude of the cross product.
The magnitude of is: Step 4: Calculating the area.
Thus, the area of the triangle is: Step 5: Conclusion.
Therefore, the area of the triangle is .

Step 1: Parametric Form of the Equation of a Line.
The given equation represents a line in 3D space. We can express the parametric form as: Thus, the parametric equations of the line are: Step 2: Vector Form.
The vector equation for the line can be written as: Where is the position vector of a point on the line, and is the direction vector. Step 3: Conclusion.
Thus, the vector equation of the line is: which corresponds to option (D).

Step 1: Understanding the Cross Product.
The cross product of any vector with itself is always 0. Therefore, Step 2: Taking the Magnitude.
The magnitude of the zero vector is always zero. Thus, Step 3: Conclusion.
Hence, the correct answer is , corresponding to option (D).