Problem 6 - Precalculus : Building Concepts and Connections

Question

Graph each of the given vectors in standard position. $$\langle-2,-5.5\rangle$$

Step-by-Step Solution

Answer

The vector \(\langle-2,-5.5\rangle\) can be graphed in standard position by drawing a line from the origin (0,0) to the point (-2,-5.5) on the Cartesian plane, with an arrow on the end to indicate direction.

Step 1: Identify the Components

Firstly, recognize the given vector components. Here, -2 is the x-component and -5.5 is the y-component of the vector \(\langle-2,-5.5\rangle\). This means the vector starts from the origin (0,0) and points to the coordinates (-2,-5.5).

Step 2: Draw the coordinate plane

On a graph, draw horizontal (x-axis) and vertical (y-axis) lines. The intersection of these lines is the origin (0,0).

Step 3: Graph the Vector

Start at the origin (0,0) and draw a line to the point (-2,-5.5) on the Cartesian plane. This line represents the vector and it's important to draw an arrow at the end of the line to show the direction of the vector. The arrow starts from origin and points towards the point (-2,-5.5).

Key Concepts

Standard Position Vectors

Understanding standard position vectors is crucial for visualizing and working with vectors in mathematics and physics. A vector in standard position is one that has its initial point at the origin of the coordinate system, which is the point \(0, 0\) in two dimensions. The terminal point of the vector is determined by its components.

For example, the vector with components \(\langle -2, -5.5 \rangle\) begins at the origin and extends to the point (-2, -5.5) on the Cartesian plane. This way of representing vectors simplifies many mathematical operations, such as vector addition and scalar multiplication. It is also helpful in determining the magnitude and direction of the vector easily.

Vector Components

The components of a vector are the projections of the vector along the axes of the coordinate system. In simpler terms, these components tell us how far to move in each direction from the origin to get to the terminal point of the vector.

The vector \(\langle -2, -5.5 \rangle\) has two components: -2 is the x-component, and -5.5 is the y-component. This tells us that from the origin, we should move 2 units left (since it's negative) and 5.5 units down (also negative) to reach the end of the vector. Understanding these components is fundamental in performing calculations involving vectors, such as computing the result when vectors are combined or finding the vector's length.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface determined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The intersection of these two axes is called the origin, from where all points on the plane are measured.

To graph the vector \(\langle -2, -5.5 \rangle\), we start by drawing the coordinate plane with both x and y-axes labeled and scaled appropriately. Beginning at the origin, we graph the vector by plotting a point at the coordinates given by its components and drawing a line from the origin to that point. In essence, the coordinate plane serves as a backdrop to graphically represent vectors and other geometric figures, making it a fundamental tool in mathematics.