Problem 10 - 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. $$\overrightarrow{O S}+\overrightarrow{S Q}$$

Step-by-Step Solution

Answer

The magnitude of the resultant vector \(\overrightarrow{OS} + \overrightarrow{SQ}\) is \(2\sqrt{13}\).

Step 1: Find Vector Representation

First, we need to find the vector representations of \(\overrightarrow{OS}\) and \(\overrightarrow{SQ}\). We find a vector from two points by subtracting the coordinates of the initial point from the terminal point. For \(\overrightarrow{OS}\), the initial point is \(O(0,0)\) and the terminal point is \(S(5,9)\). So, \(\overrightarrow{OS} = (5-0, 9-0) = (5, 9)\).For \(\overrightarrow{SQ}\), the initial point is \(S(5,9)\) and the terminal point is \(Q(4,6)\). So, \(\overrightarrow{SQ} = (4-5, 6-9) = (-1, -3)\).

Step 2: Calculate Vector Sum

Now, we calculate the sum \(\overrightarrow{OS} + \overrightarrow{SQ}\), by adding corresponding components of each vector.\[\overrightarrow{OS} + \overrightarrow{SQ} = (5, 9) + (-1, -3) = (5 + (-1), 9 + (-3)) = (4, 6)\]

Step 3: Compute Magnitude of the Resultant Vector

The magnitude of a vector \((a, b)\) is given by the formula \(\sqrt{a^2 + b^2}\).For the resultant vector \((4, 6)\), compute its magnitude:\[|\overrightarrow{OS} + \overrightarrow{SQ}| = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}\]

Step 4: Parallelogram Law Verification

According to the parallelogram law, the resultant of adding two vectors \(\overrightarrow{A}\) and \(\overrightarrow{B}\) placed tail to tail forms the diagonal of a parallelogram.Notice that \(\overrightarrow{OS}\) and \(\overrightarrow{SQ}\) can be visualized as adjacent sides of a parallelogram, and thus their sum \(\overrightarrow{OQ} = (4, 6)\) would indeed represent the diagonal of this parallelogram formed with \(O, S, Q\).

Key Concepts

Parallelogram Law

The parallelogram law is a fundamental principle for vector addition. It beautifully illustrates how two vectors can be added visually.
Here's the core idea:

• To use the parallelogram law, you place two vectors, say \(\overrightarrow{A}\) and \(\overrightarrow{B}\), so that they originate from the same point (tail to tail).
• Next, imagine each vector as a side of a parallelogram.
• The "resultant" vector, the sum of the two, is the diagonal running from the point where the vectors meet to the opposite corner of the parallelogram.

In our original problem, vectors \(\overrightarrow{OS}\) and \(\overrightarrow{SQ}\) are treated as sides of a parallelogram. Their sum, \(\overrightarrow{OQ} = (4, 6)\), represents the diagonal and thus confirms the solution. The parallelogram law is essential because it gives a quick visual depiction of vector addition, which is appreciated not only in physics but also in various fields of engineering and computer graphics.

Magnitude of a Vector

The magnitude of a vector tells us about its size or length. It is crucial for understanding the spatial distance covered by the vector, irrespective of direction.
For any vector \((a, b)\), the magnitude is computed using the formula \[\sqrt{a^2 + b^2}\]Imagine you have a vector from point O (0, 0) to S (5, 9), represented as \(\overrightarrow{OS} = (5, 9)\). The magnitude would be:

• Calculate the squares: \(5^2 = 25\) and \(9^2 = 81\).
• Sum these values: \(25 + 81 = 106\).
• Take the square root: \(\sqrt{106}\).

This process shows how the length of the vector is essentially the hypotenuse of a right triangle with sides of length \(a\) and \(b\). Understanding the magnitude helps in various practical applications, such as determining velocity, where direction is secondary, and measuring distances in coordinate geometry.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is where algebra meets geometry through graphs and coordinates. It offers an algebraic way of representing and analyzing spatial figures, crucial in performing tasks such as vector addition.
In this context:

• Any point in the plane can be represented as a coordinate pair, such as \((x, y)\).
• Vectors, then, are simply directed line segments from one point to another with both magnitude and direction.
• By defining vectors through coordinates, you can easily perform operations like addition or scaling.

Take the vector \(\overrightarrow{OS} = (5, 9)\) from our problem:

• \((5, 9)\) tells us that the vector starts at the origin \((0, 0)\) and points towards \(x = 5\), \(y = 9\).
• Changing these values will directly change the vector's orientation or starting/ending points.

Coordinate geometry simplifies solving various geometric problems, making it invaluable in both simple tasks and complex engineering design solutions.