Problem 24 - Precalculus

Question

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{Q P}+\overrightarrow{Q R}$$

Step-by-Step Solution

Answer

The sum of the vectors is \((-5, -6)\) and its magnitude is \(\sqrt{61}\).

Step 1: Identify Vectors

First, determine the components of each vector involved in the problem. The vector from point \(Q\) to point \(P\), denoted as \(\overrightarrow{QP}\), is calculated as the difference between the coordinates of \(P\) and \(Q\): \[\overrightarrow{QP} = (x_2-x_1, y_2-y_1) = ((-1)-4,3-6) = (-5,-3)\]Similarly, the vector from point \(Q\) to point \(R\), denoted as \(\overrightarrow{QR}\), is:\[\overrightarrow{QR} = (4-4,3-6) = (0,-3)\]

Step 2: Compute Vector Sum

Add the components of the vectors \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\) to find the sum \(\overrightarrow{QP} + \overrightarrow{QR}\):\[\overrightarrow{QP} + \overrightarrow{QR} = (-5, -3) + (0, -3) = (-5, -6)\]

Step 3: Calculate Magnitude

The magnitude of the resulting vector \((-5, -6)\) is calculated using the Pythagorean theorem:\[\|\overrightarrow{QP} + \overrightarrow{QR}\| = \sqrt{(-5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61}\]

Step 4: Graphical Representation (Optional)

On a graph paper, plot the points \(Q\), \(P\), and \(R\) and draw vectors \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\). To visualize the sum of these vectors using the parallelogram law, complete the parallelogram that has \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\) as its adjacent sides. The diagonal of this parallelogram represents the vector sum.

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is a branch of geometry where points are defined and plotted on a coordinate plane using ordered pairs. This method allows for precise and mathematical determination of distances and other properties.

• Points and Coordinates: In coordinate geometry, each point has a unique position determined by an ordered pair of numbers, usually written as \(x, y\). For example, the point \(P(-1,3)\) means that from the origin \(O(0,0)\), you move 1 unit to the left (since -1 is negative) and 3 units up along the y-axis.
• Vectors on the Plane: A vector such as \(\overrightarrow{QP}\) is defined by two points, where \(Q\) is the initial point and \(P\) is the terminal point. The vector's components are the differences in the respective coordinates between these two points.
• Visualizing Vectors: When drawing vectors on graph paper, it's crucial to accurately plot each point and precisely draw lines based on the vector components. This helps in analyzing vector sums graphically.

Magnitude of a Vector

The magnitude of a vector refers to its length or size. Understanding how to compute this is essential in determining how far one point is from another in the coordinate plane.

• Formula for Magnitude: The magnitude of a vector \(\overrightarrow{v} = (x, y)\) is calculated using the Pythagorean theorem. The formula is \(|\overrightarrow{v}| = \sqrt{x^2 + y^2}\).
• Step-by-Step Calculation: Let's say you have calculated a vector sum with components \((-5, -6)\). To find its magnitude, you calculate \(|\overrightarrow{QP} + \overrightarrow{QR}| = \sqrt{(-5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61}\).
• Why It Matters: Knowing the magnitude helps to understand not just direction, but the distance. In physical scenarios, it's like determining the length of a journey or the strength of a force.

Parallelogram Law for Vector Addition

The parallelogram law is a fundamental rule used for adding vectors, illustrating vector addition in a visual and intuitive manner.

• Concept Overview: According to the parallelogram law, if two vectors represent two adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram passing through the common point of the vectors.
• Practical Application: You apply the parallelogram law by drawing the vectors \(\overrightarrow{QP}\) and \(\overrightarrow{QR}\) from the same starting point on graph paper. Next, complete the parallelogram with these vectors as sides. The diagonal from the common starting point to the opposite corner represents their sum, \(\overrightarrow{QP} + \overrightarrow{QR}\).
• Benefits: This method is not only useful for understanding vector addition visually but also assists in verifying the accuracy of algebraically calculated vector sums.