Question

Find the equation of the plane passing through i-j , i+j and k.

Answer 1

Answer on Question #40013, Math, Linear Algebra

Find the equation of the plane passing through i-j, i+j and k.

Solution

Let's find the equations of the plane that pass through the points A = (1, -1, 0), B = (1, 1, 0) and C = (0, 0, 1).

$$\overline{AB} = (1 - 1, 1 - (-1), 0 - 0) = (0, 2, 0);$$ $$\overline{AC} = (0 - 1, 0 - (-1), 1 - 0) = (-1, 1, 1).$$

These points belong to the plane if the system has the solution:

$$\begin{array}{l} \overline{AX} = \lambda \cdot \overline{AB} + \mu \cdot \overline{AC}; \\ \left\{ \begin{array}{l} x - 1 = -\mu \\ y - (-1) = 2\lambda + \mu \\ z - 0 = \mu \end{array} \right. \\ \left\{ \begin{array}{ccc} x - 1 & 0 & -1 \\ y + 1 & 2 & 1 \\ z & 0 & 1 \end{array} \right. = 0 \rightarrow x + z - 1 = 0. \end{array}$$

Answer: $x + z - 1 = 0$.