Three-Dimensional Vector

Understanding Three-Dimensional Vectors

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

Notation of Three-Dimensional Vectors

Three-dimensional vectors can be notated in various ways:

1. Letter notation with an arrow above it: $\vec{a}$ or $\overrightarrow{PQ}$
2. Component notation: $(a_x, a_y, a_z)$ or $(a_1, a_2, a_3)$
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 consists of three components that represent the projection of the vector on each coordinate axis:

where:

• $a_x$ is the vector component on the x-axis
• $a_y$ is the vector component on the y-axis
• $a_z$ is the vector component on the z-axis
• $\vec{i}, \vec{j}, \vec{k}$ 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:

Example:

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

Operations on Three-Dimensional Vectors

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:

Scalar Multiplication of Vectors

Multiplying a scalar $k$ with a vector $\vec{a}$ produces a new vector with the same direction (if $k > 0$) or opposite direction (if $k < 0$) and a magnitude $|k|$ times the magnitude of $\vec{a}$.

Dot Product

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

where $\theta$ is the angle between the two vectors.

The dot product has the following properties:

1. $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ (commutative)
2. $\vec{a} \cdot \vec{b} = 0$ if and only if $\vec{a}$ and $\vec{b}$ are perpendicular (orthogonal)
3. $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Cross Product

The cross product between two vectors $\vec{a}$ and $\vec{b}$ produces a new vector $\vec{c}$ that is perpendicular to both vectors.

The magnitude of the cross product is:

where $\theta$ is the angle between the two vectors.

Applications of Three-Dimensional Vectors

Three-dimensional vectors have many applications in various fields:

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