Problem 11 - University Calculus: Early Transcendentals

Question

Find the component form of the vector. The vector from the point \(A=(2,3)\) to the origin.

Step-by-Step Solution

Answer

The component form of the vector is \((-2, -3)\).

Step 1: Understanding the problem

The problem asks us to find the component form of a vector. The vector is from a given point, in this case, point \(A=(2,3)\), to the origin \((0,0)\). The component form of a vector \(\mathbf{v}\) from point \(A = (x_1, y_1)\) to point \(B = (x_2, y_2)\) is given by \(\mathbf{v} = (x_2 - x_1, y_2 - y_1)\).

Step 2: Set coordinates for points A and origin

For point \(A\), we have the coordinates \((x_1, y_1) = (2, 3)\). For the origin \(B\), the coordinates are \((x_2, y_2) = (0, 0)\).

Step 3: Calculate the components of the vector

Using the formula \(\mathbf{v} = (x_2 - x_1, y_2 - y_1)\), substitute the values for \(x_1, y_1, x_2,\) and \(y_2\). This gives us \(\mathbf{v} = (0 - 2, 0 - 3) = (-2, -3)\).

Key Concepts

Coordinate System

A coordinate system is a method used to uniquely identify points in a space using numbers. In two dimensions, this is often done using an x-axis and a y-axis. These axes intersect at a point called the origin, typically labeled as \(0,0\). This system allows us to easily describe the position of vectors and points on a plane.
When working with vectors, each point on the plane has a unique pair of coordinates represented as \(x,y\). Here, \(x\) is the horizontal position and \(y\) is the vertical position. By using these coordinates, we can perform calculations to find vectors between points, analyze their directions, and visualizations become much more manageable.

• Coordinates specify the location of a point.
• The origin is the central point in this system.
• Vectors can be described in component form using these coordinates.

Understanding a coordinate system is essential for analyzing vectors because it lays the foundation for calculating vector components accurately.

Vector Subtraction

Vector subtraction is a process used to determine the vector between two points. Imagine you’re traveling from one point to another; to find how far and in what direction you’re moving, you use vector subtraction.
For any two points, \(A=(x_1, y_1)\) and \(B=(x_2, y_2)\), the formula for the vector from \(A\) to \(B\) is \((x_2 - x_1, y_2 - y_1)\). This subtracts the coordinates of the starting point from the destination point. For example, from \(A=(2,3)\) to \(B=(0,0)\), the subtraction yields \((-2, -3)\).

• Subtract x-coordinates: \(x_2 - x_1\).
• Subtract y-coordinates: \(y_2 - y_1\).
• Result gives vector direction and magnitude.

This calculation helps determine how one might need to "move" in the coordinate plane to go from one point to another.

Origin in Vectors

In vector analysis, the origin is a critical reference point. It is often the starting or ending point of vectors used to describe direction and magnitude relative to other points. For our example puzzle, the end point is the origin \(O=(0,0)\).
The origin marks the center of a standard coordinate plane, dividing it into four quadrants. When vectors involve the origin, it can simplify calculations because one of the points has coordinates \(0,0\). This makes the math straightforward, as any subtraction from or to the origin will simply mirror the coordinates of the other point.

• The origin simplifies vector calculations.
• Subtraction results in flipping signs of coordinates.
• Vectors to/from origin clearly show direction.

Understanding how to work with vectors starting or ending at the origin is essential, as it frequently appears in problems involving direction and movements on planes.