Three-Dimensional Coordinates and Vectors

Definition A rectangular coordinate system in space consists of a point $O$, the origin, and three mutually perpendicular number lines through $O$, the $x$, $y$ and $z$ axes, all with the same unit of length. A point $P$ gets the coordinates $(a, b, c)$ when you reach it from $O$ by moving $a$ units parallel to the $x$ axis, then $b$ units parallel to the $y$ axis, then $c$ units parallel to the $z$ axis, with negative numbers meaning steps in the negative direction. The system is right-handed when the fingers of your right hand, curled from the positive $x$ axis toward the positive $y$ axis through a right angle, leave your thumb pointing along the positive $z$ axis. This course always uses right-handed coordinates.

Definition The three coordinate planes are

They divide space into eight octants, the regions where all three coordinates are nonzero and each coordinate has a fixed sign. The first octant is the set where $x > 0$, $y > 0$ and $z > 0$. Points that lie on a coordinate plane belong to no octant. When a later problem speaks of a solid or a surface “in the first octant”, it means the closed region $x \ge 0$, $y \ge 0$, $z \ge 0$, which for a solid adds only boundary pieces of zero volume.

Definition The distance $\lvert P_1 P_2 \rvert$ between $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is the nonnegative number whose square is

Writing $\Delta x = x_2 - x_1$, $\Delta y = y_2 - y_1$ and $\Delta z = z_2 - z_1$ for the coordinate differences, the distance $d = \lvert P_1 P_2 \rvert$ is

The formula follows from the Pythagorean theorem used twice, once in a horizontal plane for the $x$ and $y$ differences and once in a vertical plane for the result and the $z$ difference. This needs perpendicular axes with a common unit, which is part of the setup above.

Definition The sphere with center $C(h, k, l)$ and radius $r > 0$ is the set of all points whose distance from $C$ is $r$. Squaring the distance formula gives its equation

An equation $x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0$ can be put in this form by completing the square in each variable. The result is a sphere if the right side comes out positive, a single point if it is zero, and no points at all if it is negative.

Definition A vector is a quantity with a magnitude and a direction. It is represented by a directed line segment, an arrow from an initial point to a terminal point, and two arrows with the same length and the same direction represent the same vector. If an arrow representing $\mathbf{a}$ starts at the origin and ends at $(a_1, a_2, a_3)$, the numbers $a_1$, $a_2$, $a_3$ are the components of $\mathbf{a}$ and we write $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$. The vector from $A(x_1, y_1, z_1)$ to $B(x_2, y_2, z_2)$ is

The vector $\overrightarrow{OP}$ is the position vector of $P$. The zero vector $\mathbf{0} = \langle 0, 0, 0 \rangle$ has length zero and no direction.

Definition The magnitude, or length, of $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ is

It is the distance between the initial and terminal points of any arrow representing $\mathbf{a}$. It is never negative, and it is zero only for $\mathbf{a} = \mathbf{0}$.

Definition For $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$, $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$ and a real number $c$,

Geometrically, $\mathbf{a} + \mathbf{b}$ is the arrow from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$ when $\mathbf{b}$ is placed at the head of $\mathbf{a}$. The vector $c\,\mathbf{a}$ has length $\lvert c \rvert \, \lvert \mathbf{a} \rvert$ and points the same way as $\mathbf{a}$ if $c > 0$ and the opposite way if $c < 0$. Two nonzero vectors are parallel when one is a scalar multiple of the other. Subtraction is $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-1)\mathbf{b}$.

Definition A unit vector is a vector of length $1$. For any nonzero vector $\mathbf{a}$, the vector

is the unit vector in the direction of $\mathbf{a}$. The zero vector cannot be normalized, since it has no direction and dividing by $0$ is undefined.

Definition A linear combination of vectors $\mathbf{v}_1$, $\mathbf{v}_2$, $\mathbf{v}_3$ is a vector $c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3$ with real numbers $c_1$, $c_2$, $c_3$. The three vectors form a basis of three-dimensional space when every vector $\mathbf{a}$ can be written as such a combination in exactly one way, that is, for each $\mathbf{a}$ there is one and only one choice of the numbers $c_1$, $c_2$, $c_3$. The standard basis vectors are

They are unit vectors along the positive axes, mutually perpendicular, and in this order they point along the positive $x$, $y$ and $z$ axes of a right-handed system. They form a basis, because

so the combination equals $\langle a_1, a_2, a_3 \rangle$ exactly when $c_1 = a_1$, $c_2 = a_2$ and $c_3 = a_3$. The coefficients with respect to $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are just the components.

Definition A force is modeled as a vector whose magnitude is its strength, measured in newtons (N), and whose direction is the direction of the push or pull. When several forces act on the same object, their combined effect is the resultant force, their vector sum. The object is in equilibrium when the resultant is the zero vector.