Problem 15 - Precalculus Mathematics for Calculus

Question

Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$P(5,3), \quad Q(1,0)$$

Step-by-Step Solution

Answer

The component form of the vector is \(\langle -4, -3 \rangle\).

Step 1: Identify Coordinates

The coordinates of point \( P \) are \((5, 3)\) and the coordinates of point \( Q \) are \((1, 0)\).

Step 2: Use the Component Form Formula

The formula to find the component form of a vector with initial point \((x_1, y_1)\) and terminal point \((x_2, y_2)\) is given by \(\langle x_2 - x_1, y_2 - y_1 \rangle\). Here \((x_1, y_1) = (5, 3)\) and \((x_2, y_2) = (1, 0)\).

Step 3: Subtract the Coordinates

Calculate the differences: \(x_2 - x_1 = 1 - 5\) and \(y_2 - y_1 = 0 - 3\).

Step 4: Compute the Component Form

The calculations yield \(1 - 5 = -4\) and \(0 - 3 = -3\). Thus, the component form of the vector is \(\langle -4, -3 \rangle\).

Key Concepts

Component Form of a Vector

In vector mathematics, the component form provides an easy way to represent a vector using its most essential parts: the difference in the horizontal (x) direction and the difference in the vertical (y) direction. By breaking down vectors into components, it simplifies many operations and calculations in coordinate geometry.
To find the component form of a vector with an initial point \( (x_1, y_1) \) and a terminal point \( (x_2, y_2) \), you use the formula \( \langle x_2 - x_1, y_2 - y_1 \rangle \). This formula calculates how much you need to move along each axis to go from the initial point to the terminal point.
So, if you have the vector starting at \( P(5, 3) \) and ending at \( Q(1, 0) \), the component form would be found by subtracting the coordinates of \( P \) from those of \( Q \). Specifically, subtract 5 from 1 for the x-component and 3 from 0 for the y-component. The component form of the vector is \( \langle -4, -3 \rangle \). This explicitly tells us that we move left by 4 units and down by 3 units from point \( P \) to point \( Q \).

Vector Subtraction

Vector subtraction is a fundamental operation that allows us to effectively compute the displacement between two points in a plane. From an application perspective, it's like asking: "What step of moves will take me from point \( P \) to point \( Q \)?"
When subtracting vectors, or in calculating differences between the initial and terminal points' coordinates, you take the terminal point's x-coordinate and subtract the initial point's x-coordinate. You do the same for the y-coordinates. This process is visually equivalent to "sliding" over to the terminal point from the initial one.

• To find the x-component, subtract the initial x-value from the terminal x-value: \( x_2 - x_1 = 1 - 5 = -4 \).
• To find the y-component, subtract the initial y-value from the terminal y-value: \( y_2 - y_1 = 0 - 3 = -3 \).

Each resultant value represents the movement along its respective axis needed to traverse from the starting point to the end point, forming a vector in the process.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves plotting and understanding geometric figures in a coordinate plane. It merges algebra and geometry through the use of a coordinate system, enabling the calculation of distances, midpoints, and vectors like those we calculate using points \( P \) and \( Q \).
In our vector context, coordinate geometry helps illustrate the vector \( \langle -4, -3 \rangle \) as a directed line segment starting from point \( P(5, 3) \) and ending at point \( Q(1, 0) \). By visualizing points and their connections through this framework, you're equipped to understand not only the magnitude and direction but also how they tie into the geometric structure of the environment.
This method also aids in visualizing transformations, rotations, and reflections on a coordinate plane, as it gives a definitive direction and magnitude for any movement across the plane.