Mathematics > Trigonometry > Vectors

Vectors in the context of trigonometry are fundamental elements that play a crucial role in both pure and applied mathematics. A vector is a mathematical object that has both a magnitude (or length) and a direction. Unlike a scalar, which only has magnitude, vectors are represented graphically by arrows. The length of the arrow denotes the magnitude, while the direction indicates the direction of the vector.

Definition

In two-dimensional space, a vector can be represented as \(\mathbf{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}\), where \(v_x\) and \(v_y\) are the components of the vector along the x-axis and y-axis, respectively. In three-dimensional space, it can be represented as \(\mathbf{v} = \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix}\).

Vector Operations

Vectors can be added and subtracted, and they can be multiplied by scalars. The operations follow specific rules:
- Addition: The sum of two vectors is given by \(\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_x + v_x \\ u_y + v_y \end{pmatrix}\).
- Scalar Multiplication: Multiplying a vector by a scalar \(c\) yields \(c\mathbf{v} = \begin{pmatrix} c v_x \\ c v_y \end{pmatrix}\).

Magnitude and Direction

The magnitude (or length) of a vector \(\mathbf{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}\) is given by:
\[ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2} \]
The direction of the vector can be described using trigonometric functions. If \(\theta\) is the angle that the vector makes with the positive x-axis, then:
\[ \cos(\theta) = \frac{v_x}{|\mathbf{v}|} \]
\[ \sin(\theta) = \frac{v_y}{|\mathbf{v}|} \]

Dot Product

The dot product of two vectors \(\mathbf{u} = \begin{pmatrix} u_x \\ u_y \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}\) is a scalar defined as:
\[ \mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y \]
This operation is closely related to the cosine of the angle between the two vectors, as given by the formula:
\[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta) \]

Cross Product

In three-dimensional space, the cross product of two vectors \(\mathbf{u} = \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix}\) is a vector defined as:
\[ \mathbf{u} \times \mathbf{v} = \begin{pmatrix} u_y v_z - u_z v_y \\ u_z v_x - u_x v_z \\ u_x v_y - u_y v_x \end{pmatrix} \]
The magnitude of this vector is given by \( |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}| \sin(\theta) \), which represents the area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\).

Applications

Vectors are used extensively in various fields such as physics, engineering, computer science, and more. For instance, in physics, vectors are used to represent quantities like force, velocity, and acceleration, which inherently have both magnitude and direction.

In summary, vectors are indispensable tools in trigonometry and broader mathematical applications, providing a way to handle directional quantities rigorously and efficiently. Understanding vector operations and properties is critical for solving complex problems in both theoretical and applied contexts.