Problem 1 - Precalculus : Building Concepts and Connections

Question

Graph each of the given vectors in standard position. $$\langle 1,0\rangle$$

Step-by-Step Solution

Answer

The vector \(\langle1,0\rangle\) is graphed as an arrow pointing to the right along the x-axis from the origin to the point (1,0).

Step 1: Understand the vector

The given vector is \(\langle1,0\rangle\). Hence, the x-coordinate is 1 and the y-coordinate is 0.

Step 2: Set-up the coordinate system

Next, Draw an x-y coordinate system on a piece of graph paper or using a graphic software if available.

Step 3: Plot the vector

Plot the vector by starting at the origin and drawing an arrow to the point (1,0). Ensure to include arrow heads to designate direction.

Step 4: Graph Verification

This completes the graph of the vector \(\langle1,0\rangle\). Your graph should show an arrow that points directly to the right along the x-axis from the origin to the point (1,0).

Key Concepts

Understanding Vectors

Vectors are fundamental in mathematics and physics. They represent quantities with both magnitude and direction, unlike scalars, which only have magnitude. A simple way to visualize a vector is by thinking of it as an arrow, where:

• The length represents the magnitude.
• The direction indicates the way the vector is pointing.

For the vector \( \langle 1,0 \rangle \), it means that the vector has a length of 1 unit and points to the right along the x-axis. The first number (1) is the horizontal movement, and the second number (0) represents no vertical movement.
Vectors are often used in physics to represent forces, velocities, or even positions, providing a visual and mathematical way to analyze these concepts.

The Coordinate System

To graph a vector, understanding the coordinate system is essential. The coordinate system is a two-dimensional plane with two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

• The x-axis extends left and right.
• The y-axis goes up and down.
• These axes intersect at the origin, which is the point \( (0,0) \).

By using this grid, one can easily locate points. For instance, the point \((1,0)\) is found by moving 1 unit along the x-axis. No movement is required along the y-axis since the second coordinate is 0.
By combining these movements, you can plot any vector given its coordinates.

The Role of the Origin

The origin \( (0,0) \) is crucial in graphing vectors, especially when they are given in standard position. This is the starting point for any vector plotted on a coordinate system.
When we say a vector is in "standard position," it means its tail is at the origin. The vector \( \langle 1,0 \rangle \) starts at the origin and then moves 1 unit to the right. No vertical movement is involved here:

• Begin at the origin.
• Move horizontally to the right by 1 unit, landing at \((1,0)\).

The origin also acts as a reference point that allows the consistent plotting of vectors, ensuring they are comparable in both placement and direction.