Problem 6 - 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. $$\overrightarrow{P O}$$

Step-by-Step Solution

Answer

The magnitude of \( \overrightarrow{P O} \) is \( \sqrt{10} \).

Step 1: Understand Vector Coordinates

Identify the coordinates of points. For vector \( \overrightarrow{P O} \), point \( P \) is at \((-1, 3)\) and point \( O \) is at \((0, 0)\).

Step 2: Calculate Vector Components

The vector \( \overrightarrow{P O} \) is directed from \( P \) to \( O \). The components of this vector are found by subtracting coordinate-by-coordinate: \((0 - (-1), 0 - 3) = (1, -3)\).

Step 3: Compute the Magnitude of the Vector

The magnitude of vector \( \overrightarrow{P O} \) is computed using the formula \( \sqrt{x^2 + y^2} \) where \( x \) and \( y \) are the components of the vector. Thus, the magnitude is \( \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \).

Step 4: Draw the Vector on Graph Paper

Draw the vector starting at point \( P(-1, 3) \) and ending at point \( O(0, 0) \) on graph paper. Use the components \((1, -3)\) to move from \( P \) to \( O \).

Step 5: Verify Using the Parallelogram Law

In our case, adding vectors is not needed, but if it were required, the parallelogram law is used to find the resultant of two vectors by placing them such that they form adjacent sides of a parallelogram and drawing the diagonal. Here, the direct calculation suffices.

Key Concepts

Vector Components

Understanding vector components is one of the fundamental aspects of vector mathematics. A vector, by definition, has both a direction and a magnitude, and it is usually represented in terms of its components. If you think of a vector as an arrow pointing from one point to another, the components are the differences in the x and y coordinates between those two points.

For instance, in the problem, to find the vector components of \( \overrightarrow{P O} \), you start with the coordinates of points \( P(-1, 3) \) and \( O(0, 0) \). The vector components are calculated by subtracting the corresponding coordinates:

• X-component: \( 0 - (-1) = 1 \)
• Y-component: \( 0 - 3 = -3 \)

Resulting in the vector components \( (1, -3) \).

These components are essential for further vector calculations, such as finding the vector's magnitude or using the vector in other mathematical operations.

Vector Magnitude

The magnitude of a vector, often called its "length" or "size," is a measure of how long the vector is irrespective of its direction. To calculate the magnitude of a vector, use the Pythagorean Theorem, which involves its components.

The formula to find the magnitude of a vector \((x, y)\) is \( \sqrt{x^2 + y^2} \), which in layman's terms means you square each component, add them up, and then take the square root of the result.
For the vector \( \overrightarrow{P O} \), with components \((1, -3)\), the magnitude is computed as follows:

• Square each component: \(1^2 = 1\) and \((-3)^2 = 9\)
• Sum them: \(1 + 9 = 10\)
• Take the square root: \( \sqrt{10} \)

Thus, the magnitude of \( \overrightarrow{P O} \) is \( \sqrt{10} \). This numerical value represents the "distance" between points \( P \) and \( O \) along the vector.

Parallelogram Law

The Parallelogram Law is a key concept when adding vectors together. It helps visualize what happens when two vectors act at a point. Though we might not need it directly for simple vectors like \( \overrightarrow{P O} \), understanding it is crucial for more complex calculations.

According to the parallelogram law, if two vectors represent two sides of a parallelogram, their sum is given by the diagonal of the parallelogram emanating from the common vertex. In simpler terms, if vectors \( \mathbf{A} \) and \( \mathbf{B} \) are positioned tail-to-tail, their vector sum \( \mathbf{A} + \mathbf{B} \) is represented by the diagonal of the parallelogram formed.
This concept is often illustrated by drawing the two vectors as adjacent sides of a parallelogram and then drawing the diagonal as the resultant vector. It provides a visual method of vector addition that is particularly useful when dealing with multiple vectors in physics and engineering applications.
Although it is not applied in the direct calculation of the magnitude of \( \overrightarrow{P O} \), it's important to recognize how the parallelogram law can offer a concrete visual framework for understanding vector addition.