Problem 7 - 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. $$\overrightarrow{P Q}+\overrightarrow{Q S}$$

Step-by-Step Solution

Answer

The magnitude of the resultant vector is \(6\sqrt{2}\).

Step 1: Determine Vector Coordinates

First, find the vector representations of \( \overrightarrow{P Q} \) and \( \overrightarrow{Q S} \) using their point coordinates. The vector \( \overrightarrow{P Q} \) is found by subtracting the coordinates of \( P \) from \( Q \), giving \( \overrightarrow{P Q} = (4 - (-1), 6 - 3) = (5, 3) \). Similarly, the vector \( \overrightarrow{Q S} \) is found by subtracting the coordinates of \( Q \) from \( S \), resulting in \( \overrightarrow{Q S} = (5 - 4, 9 - 6) = (1, 3) \).

Step 2: Compute the Sum of Vectors

To find \( \overrightarrow{P Q} + \overrightarrow{Q S} \), add corresponding components of the vectors. We have \( \overrightarrow{P Q} + \overrightarrow{Q S} = (5 + 1, 3 + 3) = (6, 6) \).

Step 3: Calculate the Magnitude of the Resultant Vector

Now, calculate the magnitude of the resultant vector \( \overrightarrow{P Q} + \overrightarrow{Q S} = (6, 6) \) using the formula for the magnitude of a vector \( \sqrt{x^2 + y^2} \). Therefore, the magnitude is \( \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \).

Step 4: Draw the Vectors and Resultant on Graph Paper

Using graph paper, draw \( \overrightarrow{P Q} \) from \( P(-1, 3) \) to \( Q(4, 6) \), and \( \overrightarrow{Q S} \) from \( Q(4, 6) \) to \( S(5, 9) \). Then, from the origin of \( \overrightarrow{P Q} \), draw the vector \( \overrightarrow{Q S} \) as it would start from the head of \( \overrightarrow{P Q} \), forming a parallelogram. The diagonal represents the resultant vector \( \overrightarrow{P Q} + \overrightarrow{Q S} \).

Step 5: Verify with the Parallelogram Law

According to the parallelogram law, the sum of two vectors \( \overrightarrow{A} \) and \( \overrightarrow{B} \) is represented by the diagonal of the parallelogram that \( \overrightarrow{A} \) and \( \overrightarrow{B} \) form. On graph paper, the diagonal starting from the common vertex of \( \overrightarrow{P Q} \) and \( \overrightarrow{Q S} \) should match the resultant calculated in Step 3, verifying the calculation of the vector sum as \( (6, 6) \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to use a coordinate system to represent geometric figures. In this system, each point in the plane is identified by a pair of numbers called coordinates, representing the horizontal (x-axis) and vertical (y-axis) positions.
To understand vector addition in coordinate geometry, consider the points with given coordinates like P(-1, 3) and Q(4, 6). These points can be illustrated on a graph. The vector \( \overrightarrow{PQ} \) is then derived by subtracting the coordinates of point P from Q, thus forming a directed line segment from P to Q.
By applying coordinate geometry principles, adding vectors like \( \overrightarrow{PQ} \) and \( \overrightarrow{QS} \) involves simply adding their corresponding components. This provides a visual and arithmetic way to tackle vector problems efficiently.

Vectors in Mathematics

Vectors in mathematics are essential quantities defined by both a direction and a magnitude. Unlike scalars, which only have magnitude (such as temperature or mass), vectors necessitate two pieces of information: direction and magnitude.
Vectors are typically represented in a two-dimensional plane as ordered pairs, like \((x, y)\), or as components in a vector, such as \(\overrightarrow{PQ} = (5, 3)\). For vector addition, align the tail of one vector with the head of another. Using the head-to-tail method helps in understanding this visually.

• Head-to-tail addition simply requires the vector’s tails to be joined to the prior vector's head.
• This property is based on the associative and commutative laws of vector addition.

By breaking vectors down into components and using operations like addition, educators can convey the profound applications of vectors in fields like physics and engineering.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is denoted using the notation \(|\overrightarrow{v}|\), where \(\overrightarrow{v}\) is the vector. It is calculated through the Pythagorean theorem, as the vector essentially forms a right triangle with its components along the axes.
For a vector \((x, y)\), the magnitude can be calculated as \(\sqrt{x^2 + y^2}\). For example, if given \(\overrightarrow{PQ} + \overrightarrow{QS} = (6, 6)\), its magnitude is \(\sqrt{6^2 + 6^2} = 6\sqrt{2}\).
Understanding magnitude is crucial as it also denotes the relative size of physical quantities like force, velocity, and displacement. Hence, taught correctly, this concept can help learners grasp how vector magnitude underpins motion and other dynamic systems.