Problem 4 - 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. $$\overrightarrow{Q S}$$

Step-by-Step Solution

Answer

The vector \( \overrightarrow{Q S} \) is \( (1,3) \) with magnitude \( \sqrt{10} \).

Step 1: Determine Vector Components

To find the vector \( \overrightarrow{Q S} \), subtract the coordinates of point \( Q \) from the coordinates of point \( S \). \( Q(4,6) \) and \( S(5,9) \), so the components of \( \overrightarrow{Q S} \) are \( S_x - Q_x \), \( S_y - Q_y \). Calculate these to get \( (5 - 4, 9 - 6) = (1, 3). \)

Step 2: Sketch Vector

On graph paper, place points \( Q(4,6) \) and \( S(5,9) \). Draw a line from \( Q \) to \( S \). This line represents the vector \( \overrightarrow{Q S} \) with components \( (1,3). \)

Step 3: Calculate Magnitude of the Vector

The magnitude of a vector \( (a, b) \) is calculated using the formula \( \sqrt{a^2 + b^2} \). For vector \( \overrightarrow{Q S} = (1, 3) \), the magnitude is \( \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}. \)

Step 4: Final Step: Using Parallelogram Law for Vector Addition

In this case, since we only need vector \( \overrightarrow{Q S} \), we don't perform addition with the parallelogram law involving this vector. However, if vectors \( \overrightarrow{Q O} \) and \( \overrightarrow{O S} \) were to be considered as two adjacent sides of a parallelogram, \( \overrightarrow{Q S} \) would correctly sum them, displaying the parallel opposite side, \( \overrightarrow{S O}. \) This step isn't necessary for the present vector, though.

Key Concepts

Vector Components

Vector components are crucial elements in understanding how vectors are formed and operate within a coordinate system. Essentially, each vector is represented by its components along the x-axis and y-axis on a plane.

For example, to find the vector \( \overrightarrow{QS} \), we subtract the coordinates of point \( Q \) from the coordinates of point \( S \). In this instance, we have \( Q(4,6) \) and \( S(5,9) \). By subtracting these coordinates, the components of the vector \( \overrightarrow{QS} \) become \( (1, 3) \).

This method helps break down a vector into understandable parts, making it easier to perform various vector operations like addition and subtraction, which are often required in solving problems involving vectors.

Parallelogram Law

The parallelogram law is a fundamental method for determining the resultant of two vectors. When you have two vectors, this law allows you to visualize their sum by arranging them as two adjacent sides of a parallelogram.

To apply the parallelogram law, consider vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{QR} \). Imagine these vectors as sides of a parallelogram. The diagonal of the parallelogram, which starts at the same point as vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{QR} \), represents the vector sum of \( \overrightarrow{PQ} + \overrightarrow{QR} \).

Even though this problem focused solely on the vector \( \overrightarrow{QS} \), understanding the parallelogram law is critical when dealing with multiple vector addition problems or understanding physical applications such as forces.

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, is a mathematical system that uses algebra to explain geometry. It's a vital tool for accurately determining the position and movement of vectors like \( \overrightarrow{QS} \).

In coordinate geometry, points are given as ordered pairs \((x, y)\), where \(x\) is the horizontal component, and \(y\) is the vertical component. The vector \( \overrightarrow{QS} \), with components \((1, 3)\), can easily be placed and plotted in this coordinate plane.

This mathematical setting not only helps in visualizing vector operations, like additions or determining magnitudes, but also in finding distances and angles between vectors, making it an indispensable tool for anybody working with vectors.