Problem 1 - Contemporary Precalculus

Question

Find the magnitude of the vector \(\overrightarrow{P Q}\). $$P=(2,3), Q=(5,9)$$

Step-by-Step Solution

Answer

Answer: The magnitude of the vector \(\overrightarrow{P Q}\) is \(3\sqrt{5}\).

Step 1: Find the coordinates of the vector \(\overrightarrow{P Q}\)

To find the coordinates of the vector \(\overrightarrow{P Q}\), subtract the coordinates of P from the coordinates of Q. $$\overrightarrow{P Q} = Q - P = (5-2, 9-3) = (3, 6)$$

Step 2: Calculate the magnitude of the vector \(\overrightarrow{P Q}\)

Now, use the formula for the magnitude of a 2-dimensional vector: $$ ||\overrightarrow{P Q}|| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Plug in the coordinates of \(\overrightarrow{P Q}\) and solve for the magnitude: $$ ||\overrightarrow{P Q}|| = \sqrt{(3)^{2} + (6)^{2}} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$$ The magnitude of the vector \(\overrightarrow{P Q}\) is \(3\sqrt{5}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes and their properties. It combines the abstract concepts of algebra with the graphical interpretations of geometry. In coordinate geometry, points are positioned on a plane using a pair of numerical coordinates. These coordinates represent the point's distance from a set reference point, usually known as the origin.

• Origin: The origin is the point where the horizontal and vertical axes intersect, typically represented as (0,0).
• Axes: The horizontal axis is referred to as the x-axis, and the vertical axis is the y-axis.

Points are typically written as ordered pairs, \((x, y)\), where \(x\) represents the horizontal position, and \(y\) represents the vertical position. In problems involving vectors in coordinate geometry, we often calculate differences in coordinates, which lead us to important properties such as the vector's magnitude.

2-Dimensional Vectors

Vectors are mathematical quantities that express both direction and magnitude. A 2-dimensional vector, often represented as \(\overrightarrow{AB}\), relates to two points in a plane, such as \(A=(x_1, y_1)\) and \(B=(x_2, y_2)\). Vectors can be used to model real-world phenomena such as force or velocity in physics. To find the vector between two points, subtract the coordinates of the starting point from the coordinates of the endpoint. For a vector from \(P=(2,3)\) to \(Q=(5,9)\), the vector is calculated as follows:\[\overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1) = (5-2, 9-3) = (3, 6)\]This calculation provides us with the vector \((3, 6)\), indicating a shift of 3 units in the x-direction and 6 units in the y-direction.

Mathematical Formulas

Mathematical formulas serve as essential tools for solving problems in various fields. Many formulas help us convert abstract concepts into solvable equations. In the context of vectors, one crucial formula is used to determine the magnitude or length of a vector.The magnitude of a vector \((x, y)\) in two dimensions is calculated using the formula:\[||\overrightarrow{PQ}|| = \sqrt{x^2 + y^2}\]For the vector \((3, 6)\), we compute the magnitude by substituting these values into the formula:\[||\overrightarrow{PQ}|| = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}\]This process provides a step-by-step approach to finding the magnitude, which is the vector's length without regard to its direction. In this example, the vector \(\overrightarrow{PQ}\) has a magnitude of \(3\sqrt{5}\), telling us the straight-line distance between the points despite their orientation.