Problem 19 - 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. $$\overrightarrow{S Q}+\overrightarrow{R O}$$

Step-by-Step Solution

Answer

The magnitude of \(\overrightarrow{SQ} + \overrightarrow{RO}\) is \(\sqrt{61}\).

Step 1: Find Vector Components

First, find the components of vectors \(\overrightarrow{SQ}\) and \(\overrightarrow{RO}\). To find \(\overrightarrow{SQ}\), subtract the coordinates of \(S\) from \(Q\) resulting in: \( (4 - 5, 6 - 9) = (-1, -3)\). For \(\overrightarrow{RO}\), subtract the coordinates of \(R\) from \(O\) resulting in: \((0 - 4, 0 - 3) = (-4, -3)\).

Step 2: Apply the Parallelogram Law

Using the parallelogram law to find the sum of vectors \(\overrightarrow{SQ}\) and \(\overrightarrow{RO}\), we will add the components of these vectors: \((-1) + (-4, -3) + (-3) = (-1 - 4, -3 - 3) = (-5, -6)\).

Step 3: Calculate the Magnitude of the Resultant

The magnitude of the resultant vector \((\overrightarrow{SQ} + \overrightarrow{RO})\) is calculated using the formula \(\sqrt{x^2 + y^2}\). Therefore, \(\sqrt{(-5)^2 + (-6)^2} = \sqrt{25 + 36} = \sqrt{61}\).

Key Concepts

Vector Components

Vector components are incredibly useful as they help us break down complex vectors into simpler, manageable parts. To find the components of a vector, we look at its changes in the horizontal (x-axis) and vertical (y-axis) directions. This is done by subtracting the coordinates of the initial point from the terminal point.
For example, let’s consider the vector \(\overrightarrow{SQ}\). The initial and terminal points are \(S(5,9)\) and \(Q(4,6)\) respectively. To find the components, we subtract the coordinates of point \(S\) from point \(Q\):

• Horizontal component (x-direction): \(4 - 5 = -1\)
• Vertical component (y-direction): \(6 - 9 = -3\)

Thus, the vector \(\overrightarrow{SQ}\) is represented by components \((-1, -3)\). Similarly, for vector \(\overrightarrow{RO}\) with points \(R(4,3)\) and \(O(0,0)\), we find:

• Horizontal component: \(0 - 4 = -4\)
• Vertical component: \(0 - 3 = -3\)

Understanding vector components helps in analyzing vectors more easily as they relate directly to coordinate points.

Parallelogram Law

The parallelogram law is a helpful geometric rule used to find the resultant of two vectors. It states that if two vectors are represented as adjacent sides of a parallelogram, then their sum is represented by the diagonal passing through the same point.
In practice, this means you can sum up the corresponding components of the given vectors to find the resultant vector. Using the exercise, vector \(\overrightarrow{SQ}\) has components \((-1, -3)\) and vector \(\overrightarrow{RO}\) has components \((-4, -3)\). Applying the parallelogram law involves straightforward addition:

• Horizontal component sum: \(-1 + (-4) = -5\)
• Vertical component sum: \(-3 + (-3) = -6\)

Thus, the resultant vector from \(\overrightarrow{SQ} + \overrightarrow{RO}\) becomes \((-5, -6)\). This visual and analytic approach simplifies the process, making it easier to understand vector addition.

Magnitude of a Vector

Calculating the magnitude of a vector helps us determine how long the vector is, essentially telling us the vector's size or length. The formula to find the magnitude of a vector with components \((x, y)\) is \(\sqrt{x^2 + y^2}\).
So, for our resultant vector \((-5, -6)\), we substitute these values into our formula:

• Square of the x-component: \((-5)^2 = 25\)
• Square of the y-component: \((-6)^2 = 36\)

Adding these gives \(25 + 36 = 61\). To get the magnitude, we take the square root, which results in \(\sqrt{61}\).
This numerical value represents the true length of our vector in the coordinate plane. Understanding vector magnitudes is crucial, as it provides information about how vectors compare in terms of length or force.