Problem 17 - 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. $$\overrightarrow{O P}+\overrightarrow{R Q}$$

Step-by-Step Solution

Answer

The sum is \((-1, 6)\) with magnitude \(\sqrt{37}\).

Step 1: Identify the Vectors

First, we need to find the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{RQ} \) from the given coordinates. \( \overrightarrow{OP} \) represents the vector from \( O(0,0) \) to \( P(-1,3) \), so it is \( \overrightarrow{OP} = (-1 - 0, 3 - 0) = (-1, 3) \). Similarly, \( \overrightarrow{RQ} \) from \( R(4,3) \) to \( Q(4,6) \) is \( \overrightarrow{RQ} = (4 - 4, 6 - 3) = (0, 3) \).

Step 2: Compute Vector Sum

To add the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{RQ} \), we apply the component-wise sum. \( \overrightarrow{OP} = (-1, 3) \) and \( \overrightarrow{RQ} = (0, 3) \). Therefore, \( \overrightarrow{OP} + \overrightarrow{RQ} = (-1 + 0, 3 + 3) = (-1, 6) \).

Step 3: Compute Magnitude of the Resultant Vector

Now find the magnitude of the resultant vector \( \overrightarrow{OP} + \overrightarrow{RQ} = (-1, 6) \). The magnitude is calculated using the formula \( \sqrt{x^2 + y^2} \). Thus, \( \sqrt{(-1)^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37} \).

Step 4: Apply Parallelogram Law

The parallelogram law states that the sum of two vectors is equal to the diagonal of the parallelogram formed by the vectors. We've already computed the vector sum as \( (-1, 6) \), which matches what we would expect visually by the parallelogram law when vectors \( \overrightarrow{OP} \) and \( \overrightarrow{RQ} \) are placed tail-to-tail.

Key Concepts

Parallelogram Law in Vector Addition

The parallelogram law is a fundamental concept for adding two vectors visually and analytically. When two vectors are placed such that they begin from the same point, they form two adjacent sides of a parallelogram. The result of the vector addition, or the sum of the vectors, is then represented by the diagonal of the parallelogram which starts from the same point. Using our exercise as an example, we have the vectors \( \overrightarrow{OP} = (-1, 3) \) and \( \overrightarrow{RQ} = (0, 3) \). When these are placed tail-to-tail, forming a figure with these two vectors as sides, they complete a parallelogram. The diagonal of this parallelogram, starting from the same origin point, is simply the vector we calculated as \((-1, 6)\). This law helps us visualize that vector addition isn't just an abstract mathematical concept, but it also represents a geometric relationship between vectors. It can be particularly useful for solving problems involving forces, velocities, or any other directed quantities.

Magnitude of a Vector

The magnitude of a vector, often denoted as the vector's length or norm, is a measure of how long the vector is from its tail to its head. To calculate the magnitude of a vector \((x, y)\), you use the formula: \[ \text{Magnitude} = \sqrt{x^2 + y^2} \] In our exercise, once we’ve computed the sum of the vectors \( \overrightarrow{OP} + \overrightarrow{RQ} = (-1, 6) \), we need to find its magnitude. Plugging into our formula: \[ \sqrt{(-1)^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37} \] This means the length of the resultant vector is \(\sqrt{37}\). Understanding how to calculate and interpret a vector's magnitude helps to comprehend the size or intensity of the vector quantity in real-world terms, such as the speed of an object in motion.

Coordinate Geometry in Vector Analysis

Coordinate geometry, also known as analytic geometry, involves using coordinates to represent geometric figures and their attributes, proving to be invaluable when analyzing vectors. It allows us to determine vectors from points on a plane easily by examining the coordinates of these points. In vector mathematics, every vector can be associated with a direct path between two points in space. For example, in the given exercise, the coordinates are translated into vectors:

• \( \overrightarrow{OP} \) is derived from points \( O(0,0) \) to \( P(-1,3) \) as \((-1, 3)\).
• \( \overrightarrow{RQ} \) comes from \( R(4,3) \) to \( Q(4,6) \) as \((0, 3)\).

Coordinate geometry assists in performing vector operations such as addition, subtraction, and finding magnitudes. It also allows you to plot these vectors on graph paper, aiding visualization and improving understanding. It bridges the gap between algebraic and geometric representations, making complex analyses simpler and more intuitive.