Vectors in Three Dimensions

Vectors in a Space with Three Coordinate Axes

A three-dimensional vector is a quantity that has both magnitude and direction in three-dimensional space. Unlike two-dimensional vectors that exist only on a plane ( $x$ - and $y$ -axes), three-dimensional vectors exist in space with three coordinate axes ( $x$ , $y$ , and $z$ axes).

Representation of Three Dimensional Vectors

A three-dimensional vector needs a third coordinate, and several notations record it. The third coordinate fixes the position along the vertical axis, and the three numbers together decide the length and the direction.

Notation of Three Dimensional Vectors

A three-dimensional vector needs a third coordinate, and each notation writes the three coordinates in a fixed order. The list below shows the common notations.

Three-dimensional vectors can be notated in various ways:

Letter notation with an arrow above it: $\vec{a}$ or $\overrightarrow{PQ}$
Component notation: $(a_x, a_y, a_z)$ or $(a_1, a_2, a_3)$
Basis notation: $a_x\vec{i} + a_y\vec{j} + a_z\vec{k}$

Components of Three Dimensional Vectors

A vector in three-dimensional space has three scalar coordinate components. Multiplying each scalar by its axis unit vector gives the corresponding vector projection:

\vec{a} = (a_x, a_y, a_z) = a_x\vec{i} + a_y\vec{j} + a_z\vec{k}

The components in this form are:

$a_x$ is the scalar component along the $x$ -axis, while $a_x\vec{i}$ is its vector projection
$a_y$ is the scalar component along the $y$ -axis, while $a_y\vec{j}$ is its vector projection
$a_z$ is the scalar component along the $z$ -axis, while $a_z\vec{k}$ is its vector projection
$\vec{i}, \vec{j}, \vec{k}$ are the unit vectors on the $x$ , $y$ , and $z$ axes

Magnitude of Three Dimensional Vectors

The magnitude or length of a three-dimensional vector $\vec{a} = (a_x, a_y, a_z)$ is determined by the formula:

|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Example:

If $\vec{a} = (3, 4, 5)$ , then the magnitude of vector $\vec{a}$ is:

\begin{aligned} |\vec{a}| &= \sqrt{3^2 + 4^2 + 5^2} \\ &= \sqrt{9 + 16 + 25} \\ &= \sqrt{50} \\ &= 5\sqrt{2} \end{aligned}

Operations on Three Dimensional Vectors

Adding and subtracting vectors in space means handling each of the three coordinates with the same rule you used in the plane. One wrong coordinate makes the whole result wrong, so a tidy line-by-line computation pays off.

Addition, subtraction, and scalar multiplication on three-dimensional vectors follow the same component rules as in two dimensions.

Addition and Subtraction of Vectors

Addition and subtraction of three-dimensional vectors are performed by adding or subtracting the corresponding components.

If $\vec{a} = (a_x, a_y, a_z)$ and $\vec{b} = (b_x, b_y, b_z)$ , then:

\vec{a} + \vec{b} = (a_x + b_x, a_y + b_y, a_z + b_z)

Scalar Multiplication of Vectors

Multiplying a scalar $k$ by a vector $\vec{a}$ scales its magnitude by $|k|$ . If $\vec{a}$ is nonzero, the direction stays the same when $k > 0$ and reverses when $k < 0$ .

k\vec{a} = (k \cdot a_x, k \cdot a_y, k \cdot a_z)

Dot Product

The dot product between two vectors $\vec{a}$ and $\vec{b}$ produces a scalar defined as:

\begin{aligned} \vec{a} \cdot \vec{b} &= a_x b_x + a_y b_y + a_z b_z \\ &= |\vec{a}||\vec{b}|\cos\theta \end{aligned}

In this formula, $\theta$ is the angle between the two vectors.

The dot product has the following properties:

$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ (commutative)
$\vec{a} \cdot \vec{b} = 0$ if and only if $\vec{a}$ and $\vec{b}$ are orthogonal. If both are nonzero, their geometric directions are perpendicular
$\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Cross Product

The cross product between two vectors $\vec{a}$ and $\vec{b}$ produces a vector whose dot product with each factor is zero:

(\vec{a}\times\vec{b})\cdot\vec{a}=(\vec{a}\times\vec{b})\cdot\vec{b}=0

The magnitude of the cross product is:

|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta

In this formula, $\theta$ is the angle between the two vectors.

When $\vec{a}\times\vec{b}\neq\vec{0}$ , its direction is perpendicular to both factors and follows the right-hand rule from $\vec{a}$ toward $\vec{b}$ . If either factor is zero or the factors are parallel, the cross product is $\vec{0}$ and therefore has no geometric direction.

Applications of Three Dimensional Vectors

Physics: To represent force, velocity, acceleration, and momentum in three-dimensional space
Computer Graphics: To represent position and movement of objects in three-dimensional space
Robotics: To control robot movement in space
Navigation: To determine direction and distance in three-dimensional space
Mechanical Engineering: For structural analysis and fluid mechanics

Vectors in a Space with Three Coordinate Axes

Representation of Three Dimensional Vectors

Notation of Three Dimensional Vectors

A three-dimensional vector needs a third coordinate, and each notation writes the three coordinates in a fixed order. The list below shows the common notations.

Three-dimensional vectors can be notated in various ways:

Letter notation with an arrow above it: $\vec{a}$ or $\overrightarrow{PQ}$
Component notation: $(a_x, a_y, a_z)$ or $(a_1, a_2, a_3)$
Basis notation: $a_x\vec{i} + a_y\vec{j} + a_z\vec{k}$

Components of Three Dimensional Vectors

A vector in three-dimensional space has three scalar coordinate components. Multiplying each scalar by its axis unit vector gives the corresponding vector projection:

\vec{a} = (a_x, a_y, a_z) = a_x\vec{i} + a_y\vec{j} + a_z\vec{k}

The components in this form are:

$a_x$ is the scalar component along the $x$ -axis, while $a_x\vec{i}$ is its vector projection
$a_y$ is the scalar component along the $y$ -axis, while $a_y\vec{j}$ is its vector projection
$a_z$ is the scalar component along the $z$ -axis, while $a_z\vec{k}$ is its vector projection
$\vec{i}, \vec{j}, \vec{k}$ are the unit vectors on the $x$ , $y$ , and $z$ axes

Magnitude of Three Dimensional Vectors

The magnitude or length of a three-dimensional vector $\vec{a} = (a_x, a_y, a_z)$ is determined by the formula:

|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Example:

If $\vec{a} = (3, 4, 5)$ , then the magnitude of vector $\vec{a}$ is:

\begin{aligned} |\vec{a}| &= \sqrt{3^2 + 4^2 + 5^2} \\ &= \sqrt{9 + 16 + 25} \\ &= \sqrt{50} \\ &= 5\sqrt{2} \end{aligned}

Operations on Three Dimensional Vectors

Addition, subtraction, and scalar multiplication on three-dimensional vectors follow the same component rules as in two dimensions.

Addition and Subtraction of Vectors

Addition and subtraction of three-dimensional vectors are performed by adding or subtracting the corresponding components.

If $\vec{a} = (a_x, a_y, a_z)$ and $\vec{b} = (b_x, b_y, b_z)$ , then:

\vec{a} + \vec{b} = (a_x + b_x, a_y + b_y, a_z + b_z)

Scalar Multiplication of Vectors

Multiplying a scalar $k$ by a vector $\vec{a}$ scales its magnitude by $|k|$ . If $\vec{a}$ is nonzero, the direction stays the same when $k > 0$ and reverses when $k < 0$ .

k\vec{a} = (k \cdot a_x, k \cdot a_y, k \cdot a_z)

Dot Product

The dot product between two vectors $\vec{a}$ and $\vec{b}$ produces a scalar defined as:

\begin{aligned} \vec{a} \cdot \vec{b} &= a_x b_x + a_y b_y + a_z b_z \\ &= |\vec{a}||\vec{b}|\cos\theta \end{aligned}

In this formula, $\theta$ is the angle between the two vectors.

The dot product has the following properties:

$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ (commutative)
$\vec{a} \cdot \vec{b} = 0$ if and only if $\vec{a}$ and $\vec{b}$ are orthogonal. If both are nonzero, their geometric directions are perpendicular
$\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Cross Product

The cross product between two vectors $\vec{a}$ and $\vec{b}$ produces a vector whose dot product with each factor is zero:

(\vec{a}\times\vec{b})\cdot\vec{a}=(\vec{a}\times\vec{b})\cdot\vec{b}=0

The magnitude of the cross product is:

|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta

In this formula, $\theta$ is the angle between the two vectors.

Applications of Three Dimensional Vectors

Physics: To represent force, velocity, acceleration, and momentum in three-dimensional space
Computer Graphics: To represent position and movement of objects in three-dimensional space
Robotics: To control robot movement in space
Navigation: To determine direction and distance in three-dimensional space
Mechanical Engineering: For structural analysis and fluid mechanics