Find system of equations such that

Question

Find system of equations that will describe:

a) plane $M \subset \mathbb{R}^3$ passing through the points $(6,1,-3), (1,5,1), (1,8,2)$

b) line $L \subset \mathbb{R}^3$ passing through $(1,2,-1), (3,4,2)$

How to this?

Answer 1

suppose the plane through $(6, 1, -3), (1, 5, 1), (1, 8, 2)$ is $$ax + by + cz = 1 $$ i row reduced $$\pmatrix{1&5&1&|&1\\1&8&2&|&1\\6&1&-3&|&1}\to \pmatrix{1&0&0&|&-4\\0&1&0&|&2.5\\0&0&1&|&-7.5}\ $$ so that we have the plane $$-4x + 2.5y -7.5z = 1 \text{ or } 8x - 5y + 15z -2 = 0 $$

for the line through $(1,2,-1), (3,4,2)$ you can give the parametric form $$x = (1-t) + 3t, y = 2(1-t) + 3t, z = -(1-t) + 2t \text{ where $t$ is any real number.}$$

Answer 2

Hint: a plane passing through the origin is given by the equation $c \cdot x = 0$ where $c$ is a vector of coefficients. So subtract one point $x_0$ from the other two points and then find the equation for the plane that passes through those two translated points and the origin, and if it's $c \cdot x = 0$ then the equation for your original plane is $c \cdot x = c \cdot x_0$.

The same procedure can be carried out for a line, by subtracting one point $x_0$ from the other point and then finding two linearly independent equations $c_1 \cdot x = 0$ and $c_2 \cdot x = 0$ that hold for your translated point. Then the equations that define your line are $c_1 \cdot x = c_1 \cdot x_0$ and $c_2 \cdot x = c_2 \cdot x_0$.

As for finding the coefficients $c,c_1,c_2$ for the equations, think in terms of finding the kernel of a matrix.