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3D Space and Vectors

Learning Targets

You should be able to

Part 1

Part 2

Concepts / Definitions

Plotting points in 3D

Sketch gridlines in the $xy$ plane and vertically to a point to show perspective. Make sure they are parallel. Right-hand rule axis.

Axes Gridlines Gridlines and points

Graphing a plane

Plot intercepts, sketch the “triangle” which represents a plane in one octant.

Triangle graphed 3D 3D Graph

The distance $d$ between the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\)

The midpoint $M$ of line segment $PQ$ with endpoints $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is \((M_x M_y M_z) = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})\)

The equation of a sphere with point $P(x, y, z)$ on the sphere and center $(h, k, l)$ and radius $r$ is \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\)

An equation of a plane can be written as \(Ax + By + Cz + D = 0\) provided $A$, $B$, and $C$ are not all zero.

Standard Unit vectors

\[\vec{\imath} = \langle 1, 0, 0 \rangle,\ \vec{\jmath} = \langle 0, 1, 0 \rangle,\ \vec{k} = \langle 0, 0, 1 \rangle\]

Dot Product

\[\vec{u} \bullet \vec{v} = u_x v_x + u_y v_y + u_z v_z\]

The equation of a line through the point $P_0 (x_0, y_0, z_0)$ in the direction of $\vec{v} = \langle a, b, c \rangle$

Parametric form: \(x = x_0 + at, y = y_0 + bt, z = z_0 + ct\)

Vector form: \(\langle x, y, z \rangle = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle\)

Cross Product

If $\vec{u} = u_1 \vec{\imath} + u_2 \vec{\jmath} + u_3 \vec{k}$ and $v_1 \vec{\imath} + v_2 \vec{\jmath} + v_3 \vec{k}$ are two vectors in space, then the cross product is \(\vec{u} \times \vec{v} = (u_2 v_3 - u_3 v_2) \vec{\imath} + (u_3 v_1 - u_1 v_3) \vec{\jmath} + (u_1 v_2 - u_2 v_1) \vec{k}\)

Cross Product

$\vec{u} \times \vec{v}$ is a vector that is orthogonal to $\vec{u}$ and $\vec{v}$ $\lvert \vec{u} \times \vec{v} = \lvert\vec{u}\rvert \lvert\vec{v}\rvert \sin{\theta}$, where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$.
$\lvert\vec{u} \times \vec{v} \rvert$ is the area of the parallelogram having non-zero vectors $\vec{u}$ and $\vec{v}$ as adjacent sides.

Cross Area

NOTE: Direction of vector is determined by the right-hand rule (order matters).

Right Hand Rule Torque Distance Force

Torque (in physics) is a force, and is how much turning power you have. It is the cross product of the radius vector and force vector. Units are Newton-meter or lbs-ft, not same as work, which is energy, and Newton-meter is joule.
(Distance not in same direction, it’s perpendicular; multiply by unitless rotation (radians) to get work.)

The standard equation of a plane containing the point $P (x_1, y_1, z_1)$ and having nonzero normal (perpendicular) vector $\vec{n} = \langle a, b, c \rangle$ is \(a(x - x_1) + b(y - y_1) + c(z - z_1) = 0\)

The general form after being simplified is \(ax + by + cz + d = 0\)

The angle between two planes can be found by \(\cos{\theta} = \frac{n_1 \bullet n_2}{\lvert n_1\rvert \lvert n_2\rvert}\) where $n_1$ and $n_2$ are the normal vectors of the two planes.

Normal Vectors

The equation of the line of intersection of the two planes, in parametric form, can be found by using the cross product of the two normal vectors of the planes and a point on the line. The cross product vector will be parallel to both planes, and will be in the direction of the line.

The distance between a plane and a point $Q$ (that is not in the plane) is \(d = \lvert proj_n\overrightarrow{PQ}\rvert\) where $P$ is a point in the plane and $n$ is a vector normal to the plane.