Problem 23 - 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. $$\overrightarrow{R P}+\overrightarrow{R S}$$

Step-by-Step Solution

Answer

The magnitude of \( \overrightarrow{R P} + \overrightarrow{R S} \) is \( 2\sqrt{13} \).

Step 1: Understand the Vectors

First, identify the vectors involved in the expression \( \overrightarrow{R P} + \overrightarrow{R S} \). Vector \( \overrightarrow{R P} \) is from point \( R \) to point \( P \), and vector \( \overrightarrow{R S} \) is from point \( R \) to point \( S \). This means we need to calculate \( \overrightarrow{R P} = \vec{P} - \vec{R} \) and \( \overrightarrow{R S} = \vec{S} - \vec{R} \).

Step 2: Calculate Component Vectors

Find the component form of each vector. - To compute \( \overrightarrow{R P} \): \[ \overrightarrow{R P} = (P_x - R_x, P_y - R_y) = (-1 - 4, 3 - 3) = (-5, 0) \]- To compute \( \overrightarrow{R S} \): \[ \overrightarrow{R S} = (S_x - R_x, S_y - R_y) = (5 - 4, 9 - 3) = (1, 6) \]

Step 3: Add Vectors Using Components

Add the vectors \( \overrightarrow{R P} \) and \( \overrightarrow{R S} \) by adding their respective components:\[\overrightarrow{R P} + \overrightarrow{R S} = (-5, 0) + (1, 6) = (-5 + 1, 0 + 6) = (-4, 6)\]

Step 4: Calculate Magnitude of the Resultant Vector

Use the formula for the magnitude of a vector \( \overrightarrow{a} = (a_x, a_y) \), which is \( \| \overrightarrow{a} \| = \sqrt{a_x^2 + a_y^2} \). Apply this to \( \overrightarrow{R P} + \overrightarrow{R S} = (-4, 6) \):\[\begin{align*}\| \overrightarrow{R P} + \overrightarrow{R S} \| &= \sqrt{(-4)^2 + 6^2} \&= \sqrt{16 + 36} \&= \sqrt{52} \&= 2\sqrt{13}\end{align*}\]

Step 5: Use the Parallelogram Law

To visualize the sum using the parallelogram law on a graph, sketch both vectors \( \overrightarrow{R P} \) and \( \overrightarrow{R S} \) originating from point \( R \), then complete the parallelogram. The diagonal of the parallelogram starting from \( R \) represents \( \overrightarrow{R P} + \overrightarrow{R S} \). The magnitude can be verified as previously calculated.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a way of representing geometric figures like points and lines using an ordered pair of numbers called coordinates. In this system, each point is expressed with an \(x\) and \(y\) value, forming a plane called the Cartesian plane. This is essential in vector addition as it provides a straightforward method to calculate position, distance, and direction.
In our exercise, the points are given in the form of coordinates: \( P(-1,3), Q(4,6), R(4,3), S(5,9) \). For instance, the point \(R\) has coordinates of \( (4, 3) \), which means it is 4 units along the \(x\)-axis and 3 units up the \(y\)-axis. By understanding the coordinates, we can accurately determine the vectors between any two points, a critical step in calculating their sum and magnitude.

Parallelogram Law

The parallelogram law is a method of vector addition. It involves creating a parallelogram where the two vectors act as adjacent sides. The sum of the vectors is represented by the diagonal of the parallelogram that starts from the common point of the two vectors.
Here's how it works:

• Set the tail of both vectors at the same point.
• Extend lines parallel to the vectors, thus forming a parallelogram.
• The resultant vector is the diagonal that originates from the common point of the initial vectors.

This visual method helps understand vector addition in a graphic sense. It can be very beneficial for students who find it hard to comprehend algebraically. In our scenario, \( \overrightarrow{RP} + \overrightarrow{RS} \) can be visualized using this law by plotting both vectors starting from point \(R\). The diagonal of the resulting parallelogram illustrates the resultant vector, which is consistent with the computed result.

Vector Magnitude

The magnitude of a vector measures its length or size. It's a scalar quantity, which means it has no direction. To find the magnitude of a vector given in component form \( \overrightarrow{a} = (a_x, a_y) \), use the formula:

• \[ \| \overrightarrow{a} \| = \sqrt{a_x^2 + a_y^2} \]

This formula is derived from the Pythagorean theorem. It calculates the hypotenuse length of a right triangle formed by the \(x\) and \(y\) components. Consider the resultant vector \( \overrightarrow{RP} + \overrightarrow{RS} = (-4, 6) \). Its magnitude is calculated as:
\[ \| \overrightarrow{RP} + \overrightarrow{RS} \| = \sqrt{(-4)^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \]Understanding magnitude is key to quantifying how long a vector is, which is essential in many areas of physics and engineering.

Component Vectors

Component vectors break down a vector into its horizontal (\( a_x \)) and vertical (\( a_y \)) parts, making calculations easier. This decomposition helps when adding or subtracting vectors, as it lets you work with easily managed numbers.
In the exercise, we use the coordinates of points to determine vectors \( \overrightarrow{RP} \) and \( \overrightarrow{RS} \):

• \( \overrightarrow{RP} = (P_x - R_x, P_y - R_y) = (-1 - 4, 3 - 3) = (-5, 0) \)
• \( \overrightarrow{RS} = (S_x - R_x, S_y - R_y) = (5 - 4, 9 - 3) = (1, 6) \)

Now, these component forms are easy to sum up:
\( \overrightarrow{RP} + \overrightarrow{RS} = (-5 + 1, 0 + 6) = (-4, 6) \)
This process shows how breaking down vectors simplifies operations like addition and subtraction while ensuring accuracy.