Problem 14 - 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. $$\overrightarrow{Q S}+\overrightarrow{P O}$$

Step-by-Step Solution

Answer

The magnitude of the sum \( \overrightarrow{Q S} + \overrightarrow{P O} \) is 2.

Step 1: Calculate Vectors

First, we need to find the vectors \( \overrightarrow{Q S} \) and \( \overrightarrow{P O} \). The vector \( \overrightarrow{Q S} \) is determined by subtracting the coordinates of \( Q \) from \( S \): \( (5 - 4, 9 - 6) = (1, 3) \). Similarly, the vector \( \overrightarrow{P O} \) is found by subtracting \( P \) from \( O \), giving \( (0 - (-1), 0 - 3) = (1, -3) \).

Step 2: Compute Vector Sum

Using the definition of vector addition, we add the components of vectors \( \overrightarrow{Q S} \) and \( \overrightarrow{P O} \). \( (1, 3) + (1, -3) = (1+1, 3-3) = (2, 0) \).

Step 3: Magnitude of the Resulting Vector

Calculate the magnitude of the resulting vector \( (2, 0) \). The magnitude formula is given by \( \sqrt{x^2 + y^2} \). Substituting, we find the magnitude: \( \sqrt{2^2 + 0^2} = \sqrt{4} = 2 \).

Step 4: Verify with Parallelogram Law

To apply the parallelogram law, plot vectors \( \overrightarrow{Q S} \) and \( \overrightarrow{P O} \) starting from a common point. The diagonal of the resulting parallelogram should match the sum calculated, which is \( (2, 0) \). This confirms our previous calculation.

Key Concepts

Vector Addition

Vector addition involves summing two vectors to create a new vector. Imagine each vector as an arrow pointing in space, with a specific direction and magnitude (or length). When we say 'add vectors', we align them head-to-tail and find where they end up together.

Here's how it works:

• Each vector is broken down into its components, which are essentially its movements along the x and y axes.
• Given vectors \( \overrightarrow{Q S} = (1, 3) \) and \( \overrightarrow{P O} = (1, -3) \), we add corresponding components to find the resultant vector. So, \( (1 + 1, 3 + (-3)) = (2, 0) \).

This process shows how two movements in plane space join together to create a new movement path.

Magnitude of a Vector

The magnitude of a vector is a measure of its length. It's calculated using the Pythagorean Theorem, which turns the vector's components into a single number, representing its length in space.

To determine magnitude:

• Consider the vector \( (x, y) \), where \( x \) and \( y \) are its components along the x and y axes, respectively.
• The magnitude \( \| \overrightarrow{v} \| \) is calculated by the formula: \( \sqrt{x^2 + y^2} \).
• For the vector \( (2, 0) \), which is the result of adding \( \overrightarrow{Q S} \) and \( \overrightarrow{P O} \), the magnitude is: \( \sqrt{2^2 + 0^2} = \sqrt{4} = 2 \).

This confirms the direct path, or length, of the resultant vector.

Parallelogram Law

The Parallelogram Law helps visualize and confirm the addition of two vectors. It is like creating a shape—a parallelogram—where the vectors act as adjacent sides.

For using this law:

• Vectors \( \overrightarrow{Q S} \) and \( \overrightarrow{P O} \) are drawn so that they start from the same point.
• The two vectors effectively create adjacent sides of a parallelogram.
• The diagonal of this parallelogram represents the vector sum—in this example, the diagonal formed confirms the result, \( (2, 0) \).

This method visually supports the algebraic addition and gives a nice geometric interpretation.