Problem 22 - 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. $$\overrightarrow{O P}+\overrightarrow{O Q}$$

Step-by-Step Solution

Answer

The resultant vector is \((3, 9)\) with magnitude \(3\sqrt{10}\).

Step 1: Identify the Vectors

First, determine the vector representations for \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \). Since point \( O \) is the origin \((0, 0)\), the vectors are simply the coordinates of points \( P \) and \( Q \). Thus, \( \overrightarrow{OP} = (-1, 3) \) and \( \overrightarrow{OQ} = (4, 6) \).

Step 2: Calculate the Sum of Vectors

Using the definition of vector addition, add the corresponding components of vectors \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \). The sum \( \overrightarrow{OP} + \overrightarrow{OQ} = (-1 + 4, 3 + 6) = (3, 9) \).

Step 3: Use the Parallelogram Law

To visually verify, draw \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \) on a graph. Place vectors as though they are adjacent sides of a parallelogram starting from the origin. The diagonal of the parallelogram from the origin should represent the resultant vector \( (3, 9) \).

Step 4: Compute Magnitude of Resultant Vector

The magnitude of a vector \( (a, b) \) is given by the formula \( \sqrt{a^2 + b^2} \). Apply this to the resultant vector \( (3, 9) \): \( \sqrt{3^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90} = 3\sqrt{10} \).

Key Concepts

Parallelogram Law

The parallelogram law helps us visually and mathematically understand how vector addition works. When you have two vectors, say \( \overrightarrow{OP} = (-1, 3) \) and \( \overrightarrow{OQ} = (4, 6) \), think of them as sides of a parallelogram starting from the same origin point. You draw these vectors on a coordinate plane where each one starts from the origin and extends according to its coordinate values.

Imagine transforming these two vectors into adjacent sides of a parallelogram. Once they're placed like this, the diagonal that emerges from the origin point to where the other sides meet is your resultant vector. The sum of \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \) gives us \( (3,9) \), seen as the diagonal vector in this visual representation.

• This resultant vector visually demonstrates the 'true effect' of both forces or directions when combined.
• The parallelogram law not only aids in graphical addition but offers geometric insight into vector operations.

By using the parallelogram method, you ensure a correct addition of vectors both geometrically and mathematically.

Vector Magnitude

Understanding the magnitude of a vector is essential since it represents the vector's length or "size." A vector from one point to another in a coordinate system like \((3, 9)\) connects the dots on this plan.

To calculate its magnitude, employ the formula \( \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the components of the vector. Here, for \( (3, 9) \):

• Square the components: \( 3^2 = 9 \), \( 9^2 = 81 \).
• Add the squares: \( 9 + 81 = 90 \).
• Take the square root: \( \sqrt{90} \)
• Simplify further: \( \sqrt{90} = 3\sqrt{10} \).

Thus, the vector's magnitude \( \sqrt{90} \) shows not only length but also helps us in physics to represent forces' intensity.

Coordinate System

A coordinate system is a mathematical framework that allows us to determine precise locations and relationships in a plane. For this exercise, the Cartesian coordinate plane is used, comprising an \( x \)-axis (horizontal) and a \( y \)-axis (vertical).
Each point in this system is labeled \( (x, y) \), where \( x \) signifies horizontal displacement, and \( y \) denotes vertical displacement. For example, point \( P(-1, 3) \) means:\

• Move 1 unit left from the origin along the \( x \)-axis.
• Move 3 units up along the \( y \)-axis.

The origin \( O(0,0) \) is the reference point where the axes intersect.
This system is crucial when it comes to vector representation because a vector is simply a directed line segment. The position and direction of a vector like \( \overrightarrow{OP} \) or \( \overrightarrow{OQ} \) are clear, allowing us to visualize and calculate vector sums easily.