Problem 16 - Precalculus Mathematics for Calculus

Question

Express the vector with initial point \(P\) and terminal point \(Q\) in component form. $$P(-1,3), \quad Q(-6,-1)$$

Step-by-Step Solution

Answer

The component form of the vector is \((-5, -4)\).

Step 1: Identify Coordinates of Points

The initial point \( P \) has coordinates \((-1, 3)\), and the terminal point \( Q \) has coordinates \((-6, -1)\). We identify these two coordinates as \( \text{Initial Point } (x_1, y_1) = (-1, 3) \) and \( \text{Terminal Point } (x_2, y_2) = (-6, -1) \).

Step 2: Apply the Formula for Component Form

The component form of a vector \( \vec{PQ} \) is calculated using the formula \( (x_2 - x_1, y_2 - y_1) \). Substituting the coordinates of points \( P \) and \( Q \):\[ x_2 - x_1 = -6 - (-1) = -6 + 1 = -5 \] \[ y_2 - y_1 = -1 - 3 = -4 \].

Step 3: Write the Component Form

The component form of the vector with initial point \( P \) and terminal point \( Q \) is \( (-5, -4) \).

Key Concepts

Understanding Initial and Terminal Points

Vectors are crucial elements in mathematics, especially when dealing with physics or engineering problems. They have both a direction and magnitude. To express a vector, we often refer to its initial point and terminal point.

• The initial point is where the vector starts. In this case, it is the point \(P(-1, 3)\).
• The terminal point is where the vector ends, which for us is \(Q(-6, -1)\).

Understanding these points is essential because they define the vector's direction and its length or magnitude. When given these points, our job is to find the vector's component form by subtracting coordinates of the initial point from the terminal point coordinates. This information is applied in many fields, including physics for force vectors or navigation for determining directions.

Simplifying Vector Subtraction

One of the most systematic methods for finding vectors in coordinate geometry is vector subtraction. It's all about comparing how far and in which direction the terminal point is from the initial point.

• Vector subtraction helps us to know the changes along the x-axis and y-axis separately.
• The formula used is \((x_2 - x_1, y_2 - y_1)\), where \((x_1, y_1)\) are coordinates of the initial point and \((x_2, y_2)\) are coordinates of the terminal point.
• In our example, we perform \(-6 - (-1)\) for the x-component and \(-1 - 3\) for the y-component.

These steps yield the vector in component form. This operation quickly shows us how far point Q shifts from point P in both horizontal and vertical directions. It's a handy technique, allowing easy representation of vectors and understanding their impact in space.

Applying Coordinate Geometry with Vectors

The field of coordinate geometry is a wonderful tool in mathematics that allows us to visualize and work with vectors efficiently. Here, every point can be described by coordinates, making it easy to perform vector operations such as finding the component form.

• Coordinate geometry lets us break down complex geometric problems into simple arithmetic calculations using coordinates.
• This is the basis for operations like finding vectors, calculating distances, and understanding geometric shapes as algebraic equations.
• By using the rectangular coordinate system, we can easily identify position changes as vectors across the axes.

The ability to change perspectives from geometric to algebraic (and vice versa) makes coordinate geometry incredibly useful. It's especially important in physics and computer graphics, where modeling real-world scenarios in 2D or 3D environments is common. With this approach, handling vectors becomes intuitive and aligns perfectly with mathematical logic.