Problem 9 - University Calculus: Early Transcendentals

Question

Find the component form of the vector. The vector \(\overrightarrow{P Q}\), where \(P=(1,3)\) and \(Q=(2,-1)\).

Step-by-Step Solution

Answer

The vector \( \overrightarrow{PQ} \) is (1, -4).

Step 1: Understanding Vector Components

First, we need to understand that the component form of a vector \( \overrightarrow{PQ} \) can be found by subtracting the coordinates of point \( P \) from point \( Q \).

Step 2: Expressing Points and Vector Formular

Given \( P = (1, 3) \) and \( Q = (2, -1) \), the formula to find the vector \( \overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1) \) where \( (x_1, y_1) \) are the coordinates of \( P \) and \( (x_2, y_2) \) are the coordinates of \( Q \).

Step 3: Calculating the 'x' Component

Subtract the x-coordinate of \( P \) from the x-coordinate of \( Q \): \( x_2 - x_1 = 2 - 1 = 1 \).

Step 4: Calculating the 'y' Component

Subtract the y-coordinate of \( P \) from the y-coordinate of \( Q \): \( y_2 - y_1 = -1 - 3 = -4 \).

Step 5: Constructing the Vector

Combine the results of the 'x' and 'y' components to form the vector: \( \overrightarrow{PQ} = (1, -4) \).

Key Concepts

Component Form of a Vector

When we talk about a vector in component form, we're actually looking at how the vector moves from one point to another in terms of horizontal (x-axis) and vertical (y-axis) movements.
The component form of a vector tells us the following:

• The difference in the x-coordinates of the two points it connects
• The difference in the y-coordinates of those points

To find this, you subtract the coordinates of the initial point from the terminal point. For example, if we have points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \), the vector \( \overrightarrow{PQ} \) in component form would be \((x_2 - x_1, y_2 - y_1)\).
This shows exactly how far horizontally and vertically you need to move to get from the initial point to the endpoint, making it extremely useful in both geometry and physics when dealing with vectors.

Vector Subtraction

Vector subtraction is a fundamental operation in vector calculus that's used to find the component form of a vector.
In simple terms, vector subtraction is similar to regular subtraction where you subtract corresponding components of two vectors. For a vector \( \overrightarrow{PQ} \) which starts from point \(P\) and ends at point \(Q\), the subtraction comes into play as you subtract the coordinates of point \(P\) from point \(Q\).
In our example, with points \( P = (1, 3) \) and \( Q = (2, -1) \), vector subtraction gives us:

• Subtract the x-coordinates: \( 2 - 1 = 1 \)
• Subtract the y-coordinates: \( -1 - 3 = -4 \)

This subtraction calculates how each coordinate changes from Point \(P\) to Point \(Q\), effectively breaking down the movement into two separate, easily manageable changes.

Coordinate Geometry and Vectors

Coordinate geometry provides a framework in which vectors can be studied and understood more easily. In a coordinate plane, every point is described by an x-coordinate and a y-coordinate.
Vectors in coordinate geometry can be represented graphically as arrows pointing from one point to another with specified direction and magnitude.
With the help of coordinate geometry, the vectors give a clear picture of how position changes across the plane.

• A vector’s magnitude refers to its length, determined by the formula \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).
• The direction of a vector is indicated by the angle it forms with the x-axis, given by \( \theta = \tan^{-1}\left(\frac{y_2-y_1}{x_2-x_1}\right) \)

Coordinate geometry aids in directly visualizing how one point shifts to another using a vector, therefore simplifying complex spatial problems.