Welcome to learning about three D plane equations. A plane in three dimensional space can be described by a linear equation: A x plus B y plus C z plus D equals zero. The coefficients A, B, and C form the normal vector, which is perpendicular to the plane, while D determines the plane's position in space.
The normal vector components A, B, and C determine the plane's orientation in space. For example, in the equation two x plus three y plus z minus six equals zero, the normal vector is two, three, one. This vector is perpendicular to every line that lies within the plane, which is a fundamental property of plane geometry.
The constant D in the plane equation determines how far the plane is from the origin. The distance formula is the absolute value of D divided by the square root of A squared plus B squared plus C squared. For example, x plus y plus z minus three equals zero and x plus y plus z plus three equals zero represent parallel planes with the same normal vector but different distances from the origin.
To find the equation of a plane, you need two key pieces of information: a point that lies on the plane, and the normal vector. The general formula is A times x minus x naught plus B times y minus y naught plus C times z minus z naught equals zero. Here we show an example with point one, two, one and normal vector two, one, one.
To summarize what we've learned about three D plane equations: A plane is described by A x plus B y plus C z plus D equals zero, where A, B, C form the normal vector that's perpendicular to the plane, and D determines the distance from the origin. Understanding these components helps us work with planes in three dimensional space.