Grade 12 Mathematics: Analytical Solid Geometry Flashcards

This set covers the fundamental concepts of 3D rectangular coordinate systems, including plane equations, line equations, distance formulas, and directed values.

Origin O

The fixed point in space through which three mutually perpendicular lines (the $x$-axis, the $y$-axis, and the $z$-axis) pass.

Ordered triple

An association of three real numbers $(x, y, z)$ used to determine the coordinates of a point $P$ in three-dimensional space.

xy-plane

The plane consisting of the $x$-axis and the $y$-axis, where the $z$-axis is perpendicular to it and points are of the form $(x, y, 0)$.

yz-plane

The plane consisting of the $y$-axis and the $z$-axis, where the $x$-axis is perpendicular to it and points are of the form $(0, y, z)$.

zx-plane

The plane consisting of the $x$-axis and the $z$-axis, where the $y$-axis is perpendicular to it and points are of the form $(x, 0, z)$.

Equation of the xy-plane

$$z = 0$$

Equation of the yz-plane

$$x = 0$$

Equation of the zx-plane

$$y = 0$$

Equation of a plane parallel to the xy-plane

$z = c$, where the coordinates of points on that plane are of the form $(x, y, c)$.

Equation of a plane parallel to the yz-plane

$x = a$, where the coordinates of points on that plane are of the form $(a, y, z)$.

Equation of a plane parallel to the zx-plane

$y = b$, where the coordinates of points on that plane are of the form $(x, b, z)$.

Equation of a line perpendicular to the xy-plane

For a line passing through point $(a, b, c)$, the equation is $x = a, y = b$, and coordinates on the line are of the form $(a, b, z)$.

Equation of a line perpendicular to the yz-plane

For a line passing through point $(a, b, c)$, the equation is $y = b, z = c$, and coordinates on the line are of the form $(x, b, c)$.

Equation of a line perpendicular to the zx-plane

For a line passing through point $(a, b, c)$, the equation is $x = a, z = c$, and coordinates on the line are of the form $(a, y, c)$.

Equation of the x-axis

$$y = 0, z = 0$$

Equation of the y-axis

$$x = 0, z = 0$$

Equation of the z-axis

$$x = 0, y = 0$$

Distance formula between two points

$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ for points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$.

Directed values of a line segment PQ

The values $(l, m, n)$ defined by $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$ where $P$ is $(x_1, y_1, z_1)$ and $Q$ is $(x_2, y_2, z_2)$.

Parametric equations of a point R(x, y, z) on line PQ

$x = x_1 + kl$, $y = y_1 + km$, and $z = z_1 + kn$, where $k$ is a real number and $(l, m, n)$ are the directed values of the segment $PQ$.