Problem 14 - Algebra and Trigonometry

Question

\(9-18\) . Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$ P(1,1), \quad Q(9,9) $$

Step-by-Step Solution

Answer

The vector from \( P \) to \( Q \) is \( \langle 8, 8 \rangle \).

Step 1: Understanding the Problem

We are given two points, \( P(1,1) \) and \( Q(9,9) \), and we need to find the vector that represents the direction and distance from point \( P \) to point \( Q \) in component form.

Step 2: Calculate the Change in Coordinates

To express the vector in component form, we subtract the coordinates of point \( P \) from those of point \( Q \). This gives us the changes in the x and y coordinates:\[\Delta x = Q_x - P_x = 9 - 1 = 8\]\[\Delta y = Q_y - P_y = 9 - 1 = 8\]

Step 3: Write the Vector in Component Form

Now that we have \( \Delta x = 8 \) and \( \Delta y = 8 \), we can express the vector from \( P \) to \( Q \) in component form as:\[ \langle 8, 8 \rangle \]. This vector represents the horizontal and vertical distances from \( P \) to \( Q \).

Key Concepts

Component Form

The component form of a vector is a precise way to describe a vector's direction and magnitude using coordinates. When dealing with vectors in a plane, the component form offers an easy method to express how far and in which direction a point should move from one location to another.
In the given exercise, you are asked to find the vector from point \(P(1,1)\) to \(Q(9,9)\). To do this, you calculate the change in the x-coordinates and the y-coordinates separately. This change is often termed as \(\Delta x\) and \(\Delta y\).
For any two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), the component form of the vector \(\overrightarrow{PQ}\) is given by:

• \(\Delta x = x_2 - x_1\)
• \(\Delta y = y_2 - y_1\)

Thus, the component form is expressed as \(\langle \Delta x, \Delta y \rangle\). This format is beneficial because it succinctly captures both the direction and distance covered, transforming a vector into a simple algebraic representation.

Coordinate Geometry

Coordinate geometry serves as the foundation for understanding vectors and their properties in a plane. By using a coordinate system, we can efficiently locate points, determine distances, and describe geometric figures.
The fundamental idea here is to use coordinates to describe every relevant aspect of a vector. With the points given in our problem, \(P(1,1)\) and \(Q(9,9)\), coordinate geometry enables us to determine not just the point locations but also how exactly they are related in terms of a vector.
In this exercise, coordinate geometry allows us to pinpoint the start at \(P\) and the end at \(Q\), then calculate the journey between these points using simple arithmetic operations.

• By employing subtraction of coordinates, as shown in the previous section, we can find the vector's component form.
• Coordinate geometry simplifies this representation to \(\langle 8, 8 \rangle\) in our particular example, providing clarity and efficiency in mathematical description.

Vector Subtraction

Vector subtraction is a crucial operation in vector mathematics, used to find a vector that represents the displacement from one point to another. In essence, it shows how one vector differs from another in terms of both direction and magnitude.
When you need the vector from point \(P\) to point \(Q\), you essentially subtract vector \(\overrightarrow{OP}\) from vector \(\overrightarrow{OQ}\), where \(O\) is the origin.
The subtraction formula for vectors \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is:

• The result of subtracting \(P\) from \(Q\) is \(\langle x_2 - x_1, y_2 - y_1 \rangle\).

This process reflects the movement needed to get from the initial point to the terminal point. In our case, it explains the transition from \(P(1,1)\) to \(Q(9,9)\), depicting the path traversed and the vector \(\langle 8, 8 \rangle\) as the result of this subtraction operation.