Question

The equation of a plane passing through three non-collinear points is determined using:

• A. Vector form
• B. Determinant method
• C. Cartesian equation
• D. All of the above

Hint:

If three points are collinear, the determinant will equal zero for any point $(x, y, z)$, meaning an infinite number of planes could pass through them (like pages in a book). Always ensure points are non-collinear!

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A plane is uniquely determined if three non-collinear points on it are known. Non-collinear means the points do not lie on a single straight line. There are several mathematical representations used to find this equation.

Step 2: Key Formula or Approach
If the points are $A(\vec{a})$, $B(\vec{b})$, and $C(\vec{c})$: 1. Vector form: $(\vec{r} - \vec{a}) \cdot [(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})] = 0$. 2. Determinant method: \[ \begin{vmatrix} x - x_1 & y - y_1 & z - z_1 x_2 - x_1 & y_2 - y_1 & z_2 - z_1 x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0 \] 3. Cartesian equation: $Ax + By + Cz + D = 0$.

Step 3: Detailed Explanation
All three methods are valid and interchangeable: - The Vector form uses the cross product of two vectors in the plane to find the normal vector. - The Determinant method is the most common manual calculation tool in coordinate geometry. - The Cartesian equation is the final expanded result derived from either of the first two methods.

Step 4: Final Answer
Since all forms can be used to determine the plane, the correct choice is (D).