Problem 4 - Precalculus : Building Concepts and Connections

Question

Graph each of the given vectors in standard position. $$\left\langle 0, \frac{1}{2}\right\rangle$$

Step-by-Step Solution

Answer

The given vector \(\langle 0, \frac{1}{2}\rangle\) is plotted by starting at the origin point and drawing an arrow to the point (0, \(\frac{1}{2}\)). As result we have a line that goes from the origin straight upwards to the point \((0,\frac{1}{2})\).

Step 1: Understand the Vector Components

Here, the vector \(\langle 0, \frac{1}{2}\rangle\) is presented. It means that the vector has its 'x' component as 0 and 'y' component as \(\frac{1}{2}\). Remember that in a 2D coordinate system, the first component represents 'x', the horizontal direction, and the second represents 'y', the vertical direction.

Step 2: Plot the Vector

Begin at the origin point (0,0). Here, since the 'x' component is 0, there is no movement in the horizontal direction. The 'y' component is \(\frac{1}{2}\), meaning we move half a unit in the positive vertical direction. So, place the tip of the vector at the point (0, \(\frac{1}{2}\)). Note that the vector is drawn as an arrow originating from the origin point (0,0) and ending at the point (0, \(\frac{1}{2}\)).

Key Concepts

Graphing Vectors

Graphing vectors is a visually intuitive way to represent them on a coordinate plane. Vectors are typically represented by arrows, which makes it easy to see both the direction and magnitude. To graph a vector, you begin by marking its initial point, which is usually the origin \((0, 0)\) in the standard position. The endpoint is determined by the vector's components.

• The direction of the arrow indicates the vector's direction, defined by the order of its components.
• The length of the arrow is proportional to the vector's magnitude, which can be calculated using the Pythagorean theorem if needed.
• When graphing, always ensure that your scale on both axes is consistent to accurately portray the vector's magnitude.

Graphing provides a clear, visual portrayal of vector components and their relative influences on the vector's overall direction and magnitude. For instance, the vector \(\langle 0, \frac{1}{2}\rangle\) plotted from the origin will form an arrow pointing straight up along the y-axis, showcasing purely vertical movement with no horizontal shift.

Two-Dimensional Coordinate System

A two-dimensional coordinate system is the framework used for graphing vectors. It allows us to represent both the horizontal and vertical movements on a plane. The most common model is the Cartesian coordinate system, structured with an \(x\)-axis and a \(y\)-axis, intersecting at the origin \((0, 0)\).

• The \(x\)-axis runs horizontally, allowing for left-right movement.
• The \(y\)-axis runs vertically, providing up-down movement.
• Any point in this system is given as \((x, y)\), where \(x\) is the horizontal component, and \(y\) is the vertical component.
• Vectors are not limited to starting from the origin, but the standard position means they do, simplifying calculations and visualizations.

This setup is crucial for graphing because it provides a straightforward reference for measuring movement influenced by each component of the vector separately. The vector \(\langle 0, \frac{1}{2}\rangle\) is an excellent example within this system, as it illustrates pure movement along the y-axis.

Vector Components

Understanding vector components is crucial for interpreting vectors correctly. A vector is defined by its components, which describe its direction and magnitude. In a two-dimensional system, a vector \(\langle a, b\rangle\) has two components:

• The first component \(a\) is the horizontal or \(x\) component. It tells you how far left or right to move.
• The second component \(b\) is the vertical or \(y\) component. It tells you how far up or down to move.

These components are used to plot the vector on a graph. By starting at the initial point (usually the origin), you move according to the components:

• The \(x\)-component instructs horizontal movement.
• The \(y\)-component dictates vertical movement.
• For the vector \(\langle 0, \frac{1}{2}\rangle\), the absence of a horizontal component means no left or right movement, so focus is solely on moving \(\frac{1}{2}\) unit upward.

This separation of influences into \(x\) and \(y\) parts is what makes manipulating and understanding vectors particularly useful in physics and engineering applications.