Problem 15 - 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. $$\overrightarrow{S R}+\overrightarrow{P O}$$

Step-by-Step Solution

Answer

The sum of vectors is \( (0, -9) \) with a magnitude of 9.

Step 1: Find Vector Components

Calculate the components of vectors \( \overrightarrow{SR} \) and \( \overrightarrow{PO} \) using the coordinates of the points. For \( \overrightarrow{SR} \), use the formula \((x_2-x_1, y_2-y_1)\). Thus, \( \overrightarrow{SR} = (4-5, 3-9) = (-1,-6) \). For \( \overrightarrow{PO} \), use the same formula: \( \overrightarrow{PO} = (0+1, 0-3) = (1,-3) \).

Step 2: Compute Vector Sum

Add the vectors \( \overrightarrow{SR} \) and \( \overrightarrow{PO} \). The sum is found by adding the corresponding components: \((-1, -6) + (1, -3) = ((-1+1), (-6-3)) = (0, -9) \).

Step 3: Calculate Magnitude of the Sum

The magnitude of a vector \( (a, b) \) is \( \sqrt{a^2 + b^2} \). For the vector \( (0, -9) \), the magnitude is \( \sqrt{0^2 + (-9)^2} = \sqrt{81} = 9 \).

Step 4: Apply the Parallelogram Law (Optional Verification)

Although we've already computed the sum, the parallelogram law can confirm the result. Place vectors \( \overrightarrow{SR} \) and \( \overrightarrow{PO} \) tail-to-tail and complete the parallelogram. The resultant vector \( (0, -9) \) is the diagonal of the parallelogram, matching our computed sum.

Key Concepts

Vector Components

In the context of vector addition, understanding vector components is essential. To find the components of a vector, you start by looking at its initial and terminal points. A vector in 2D space can be represented as \((x_2 - x_1, y_2 - y_1)\), where \((x_1, y_1)\) are the coordinates of the starting point and \((x_2, y_2)\) the ending point.
For example, consider vector \(\overrightarrow{SR}\) with points \(S(5, 9)\) and \(R(4, 3)\). Its components are calculated as:
\[\overrightarrow{SR} = (4 - 5, 3 - 9) = (-1, -6)\]
Similarly, for vector \(\overrightarrow{PO}\) with points \(P(-1, 3)\) and \(O(0, 0)\), the components are:
\[\overrightarrow{PO} = (0 + 1, 0 - 3) = (1, -3)\]
This breakdown helps in visualizing and performing further operations on vectors, such as addition.

Magnitude of a Vector

The magnitude of a vector gives its length, which is a crucial aspect when dealing with vectors in coordinate geometry. To find the magnitude of a vector \((a, b)\), we use the formula \(\sqrt{a^2 + b^2}\).
This formula applies to any two-dimensional vector, allowing us to calculate its length geometrically.
Consider the vector \((0, -9)\), derived from the sum of \(\overrightarrow{SR}\) and \(\overrightarrow{PO}\). Its magnitude is:
\[\sqrt{0^2 + (-9)^2} = \sqrt{81} = 9\] This calculation shows the vector's length from the origin to its endpoint. Magnitude is essential for understanding vector direction and comparing vector sizes.

Parallelogram Law

The parallelogram law is a fundamental concept for understanding vector addition. It states that when two vectors are represented as adjacent sides of a parallelogram, their sum can be found as the diagonal passing through their common tail.
This law provides a geometric method of verifying vector addition.
For the vectors \(\overrightarrow{SR}\) and \(\overrightarrow{PO}\), arranging them tail-to-tail allows us to complete the parallelogram.
The diagonal of this parallelogram represents the addition of these vectors. For this example, the resultant diagonal is \( (0, -9)\), confirming our computed vector sum from a geometric perspective.
You can visualize vector addition through this approach, appreciating the relationship between geometry and algebra.

Coordinate Geometry

Coordinate geometry intertwines algebra and geometry, providing tools to analyze vector behavior within a coordinate plane. It allows us to calculate distances, slopes, and other properties essential for vector operations.
Using the coordinates of points, we dissect vectors into components and articulate their interactions.
In this exercise, each point like \(P(-1, 3)\), \(Q(4, 6)\), is plotted on the plane to visualize vectors \(\overrightarrow{SR}\) and \(\overrightarrow{PO}\).
Through coordinate geometry, we calculate these vectors' sums, magnitudes, and verify using the parallelogram law. It's the backbone of vector application, marrying algebraic formulas with geometric interpretations to deepen our understanding.