Problem 11 - Thomas Calculus

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 \(\langle -2, -3 \rangle\).

Step 1: Understanding the Problem

The problem asks us to find the component form of the vector from the point \(A = (2, 3)\) to the origin, which is \((0, 0)\). Vectors in component form are represented with their horizontal and vertical components.

Step 2: Calculating the Horizontal Component

The horizontal component of a vector is found by subtracting the x-coordinate of the initial point from the x-coordinate of the terminal point. For our vector, it starts at \((2, 3)\) and ends at the origin \((0, 0)\). So, the horizontal component is: \[ x_2 - x_1 = 0 - 2 = -2 \]

Step 3: Calculating the Vertical Component

The vertical component of a vector is found by subtracting the y-coordinate of the initial point from the y-coordinate of the terminal point. Using the points \((2, 3)\) as the start and \((0, 0)\) as the end, we calculate: \[ y_2 - y_1 = 0 - 3 = -3 \]

Step 4: Writing the Component Form

Once we have both the horizontal and vertical components, we can write the vector in component form as: \[ \langle -2, -3 \rangle \] This vector describes moving 2 units left and 3 units down from the point \((2, 3)\) to the origin.

Key Concepts

Vector Mathematics

Vector mathematics is all about dealing with quantities that have both magnitude and direction. These quantities are known as vectors. Vectors are incredibly useful in a variety of fields including physics, engineering, and computer science.
A key aspect of vector mathematics is being able to represent a vector in component form. This form breaks down the vector into horizontal and vertical parts, which correspond to directions in a 2-dimensional coordinate system.

• The horizontal component represents movement along the x-axis.
• The vertical component represents movement along the y-axis.

Understanding vectors this way allows for easier calculations and operations, such as addition, subtraction, and scaling of vectors.
Component representation also makes it easier to visualize the direction and length of vectors, aiding both in theoretical and practical applications of vector mathematics.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use a coordinate system to study the positions of points and geometric figures.
In this system, points are expressed as coordinates like \(x, y\) in a 2-dimensional plane. This allows for precise definitions and calculations regarding distances and directions between various points.
When working with vectors within coordinate geometry, each point can serve as either the initial or terminal point.

• Vectors can be created to show movement from one point to another.
• By subtracting the coordinates of the starting point from the ending point, we find the vector components.

This approach gives us a clear picture of how vectors behave in a plane, and is essential for tasks such as vector addition and subtraction – turning abstract spatial concepts into numerical problems that can be solved logically.

Vector Subtraction

Vector subtraction is a fundamental operation in vector mathematics. It involves finding the difference between two vectors, which can represent such things as changes in position or displacement.
The operation of vector subtraction occurs component-wise:

• The horizontal component changes by subtracting the initial vector's x-coordinate from the terminal vector's x-coordinate.
• The vertical component changes by subtracting the initial vector's y-coordinate from the terminal vector's y-coordinate.

Consider the example of finding a vector from point \(A = (2, 3)\) to the origin \(0, 0\):

• The horizontal subtraction would be \(0 - 2 = -2\).
• The vertical subtraction would be \(0 - 3 = -3\).

The resulting vector \(\langle -2, -3 \rangle\) clearly represents the shift needed to move from the point \(A\) to the origin.
By understanding vector subtraction, you gain a more intuitive sense of shifting and displacement in the coordinate plane, which is crucial for more advanced studies and applications.