Question

If \(\mathbf{v}=\langle a, b\rangle\) is an algebraic vector whose initial point is the origin, then \(\mathbf{v}\) is called a(n) _____ vector.

Short answer

Position vector

Step-by-step solution

Understand the Concept of Vectors

A vector is a mathematical object that has both a magnitude (or length) and a direction. It is often represented as an arrow pointing from one point to another.

Identify the Initial Point

The initial point of the vector \(\mathbf{v}\) given in the problem is the origin. The origin in a coordinate system is the point \( (0, 0) \).

Determine the Type of Vector

When a vector's initial point is the origin, it is often referred to as a position vector. This is because it points to a position in space from the origin.

Conclude the Answer

Since the initial point of the vector provided is the origin, \(\mathbf{v} = \langle a, b \rangle\) is called a position vector.

Key concepts

Vectors

A vector is a fundamental concept in mathematics and physics. It is an entity that possesses both magnitude and direction. Think of a vector as an arrow: the length of the arrow represents the magnitude, and where the arrow points signifies the direction.

Vectors are commonly denoted by bold letters or by letters with arrows on top, like \(\text{\mathbf{v}}\). In 2D space, a vector can be represented as \(\mathbf{v} = \langle a, b \rangle\), where \(\text{\langle a, b \rangle}\rangle\) are its components along the x- and y-axes, respectively.

Understanding vectors is crucial since they are used to represent various physical quantities, such as velocity, force, and displacement.

• Magnitude: The length or size of the vector.
• Direction: Where the vector points.

Magnitude and Direction

The magnitude and direction of a vector are two key characteristics that define it completely.

• Magnitude: The magnitude of a vector is its length. In 2D space, you can find the magnitude using the Pythagorean theorem: \(|\mathbf{v}| = \sqrt{a^2 + b^2}\|\). This formula comes from the right triangle formed by the components of the vector.
• Direction: The direction of a vector is given by the angle it makes with a reference axis, usually the x-axis. The direction can be found using trigonometric functions. Specifically, the angle \(\theta\) can be found using \(\tan^{-1}(\frac{b}{a})\).

Understanding these properties helps in visualizing the vector in space and solving problems related to vector analysis.

For example, if you have a vector \(\text{\mathbf{v}} = \langle 3, 4 \rangle\), you can find its magnitude as follows:
\(|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

Similarly, you can determine its direction by calculating the angle with the x-axis.

Coordinate System

A coordinate system is essential for describing the position of vectors. The most common coordinate system used is the Cartesian coordinate system, defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

In 2D space, any point can be represented as \( (x, y) \). For vectors, this point signifies the end point when the initial point is the origin \( (0, 0) \).

For example, the vector \(\text{\mathbf{v}} = \langle a, b \rangle\) means it starts at \( (0, 0) \) and ends at \( (a, b) \). This is called a position vector because it specifies the exact position of the point \( (a, b) \) from the origin.

Coordinate systems are used to break vectors into components, making it easier to analyze and solve various problems in geometry, physics, and engineering.

• Origin: The point \((0, 0)\) where both axes intersect.
• Axes: The reference lines (x-axis and y-axis) used to define positions.
• Coordinates: The \( (x, y) \) values that describe a point in the system.