Problem 5 - 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. $$\overrightarrow{O P}$$

Step-by-Step Solution

Answer

Magnitude of \(\overrightarrow{OP}\) is \(\sqrt{10}\) with direction from \(O\) to \((-1, 3)\).

Step 1: Identify the Vectors

The exercise asks us to draw and calculate the magnitude of the vector \(\overrightarrow{OP}\), where \(O\) is the origin \((0,0)\) and Point \(P\) has coordinates \((-1, 3)\). Thus, the vector \(\overrightarrow{OP}\) is described by the position change from \(O\) to \(P\), which is \((-1, 3)\).

Step 2: Draw the Vector

On a graph paper, plot the point \(O\) at the origin \((0, 0)\) and point \(P\) at coordinates \((-1, 3)\). Draw a directed line segment from the origin to \(P\) to represent the vector \(\overrightarrow{OP}\).

Step 3: Compute the Magnitude of \(\overrightarrow{OP}\)

The magnitude of a vector \(\overrightarrow{OP} = (x, y)\) is given by the formula \(\sqrt{x^2 + y^2}\). For \(\overrightarrow{OP} = (-1, 3)\), calculate:\[\magnitude = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10}\]

Step 4: Interpret the Results

The vector \(\overrightarrow{OP}\) represents a translation from the origin to point \((-1, 3)\), with a magnitude of \(\sqrt{10}\). This vector and its magnitude demonstrate the distance and direction from the origin to point \(P\).

Key Concepts

Vector Magnitude

When working with vectors, a key concept to grasp is the magnitude, which effectively measures the length or size of a vector. Imagine it as the distance from the starting point to the endpoint of the vector.

In a Cartesian coordinate system, a vector such as \( \overrightarrow{OP} \) can be described using its components \( (x, y) \). The magnitude of a vector \( \overrightarrow{OP} = (x, y) \) is derived through the Pythagorean theorem. This is done using the formula:

\[ \text{Magnitude} = \sqrt{x^2 + y^2} \]

For instance, the vector \( \overrightarrow{OP} \) with components \((-1, 3)\), has a magnitude calculated as follows:

• Square each component: \((-1)^2 = 1\) and \((3)^2 = 9\).
• Add the squares: \(1 + 9 = 10\).
• Take the square root of the sum: \(\sqrt{10}\).

The result, \(\sqrt{10}\), indicates both the direction and length of the movement from point \(O\) to point \(P\). Understanding vector magnitude allows you to quantify how far and in what direction a point moves in a coordinate plane.

Coordinate Geometry

Coordinate geometry, or Cartesian geometry, is the study of geometric figures using a coordinate system. This system allows us to describe the position and relationship of points that define shapes, such as lines and curves.

Through coordinate geometry, points like \(O(0,0)\) and \(P(-1,3)\) are represented in a two-dimensional plane where each point corresponds to an \(x\)- and \(y\)-coordinate, respectively.

This approach simplifies how we draw vectors and calculate properties such as distance, slope, and position. For instance, the vector \( \overrightarrow{OP} \) is depicted by plotting point \(O\) at the origin and point \(P\) at \(-1,3\), effectively demonstrating the vector's path and direction.

• Easy visualization: By having coordinates, we can easily plot and visualize geometric relations on graph paper.
• Precise calculations: Using coordinates, we can precisely calculate distances, midpoint, and other properties using algebraic formulas.

Understanding coordinate geometry is essential in translating mathematical problems into visual formats, making complex concepts more tangible and simpler to solve.

Parallelogram Law

The parallelogram law is a principle used to determine the resultant of two vectors. When two vectors originate from the same point, their resultant can be found using this rule. Let's consider vectors \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\).

According to the parallelogram law, a parallelogram is formed by using the vectors as adjacent sides. The diagonal of this parallelogram starting from the common origin gives the resultant vector.

This can be visually represented and executed as follows:

• Draw both vectors, \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\), starting from the same point.
• Complete the parallelogram by drawing lines parallel to these vectors.
• The diagonal from the origin of the parallelogram represents the vector sum \(\overrightarrow{OA} + \overrightarrow{OB}\).

This law simplifies the graphical representation of vector addition and is instrumental in fields such as physics and engineering for calculating forces, velocities, and more. Understanding the parallelogram law assists in visualizing how vectors combine, thereby making vector addition intuitive and manageable.