Problem 3 - Contemporary Precalculus

Question

Find the magnitude of the vector \(\overrightarrow{P Q}\). $$P=(-7,0), Q=(-4,-5)$$

Step-by-Step Solution

Answer

Question: Find the magnitude of the vector \(\overrightarrow{P Q}\) if point P has coordinates (-7, 0) and point Q has coordinates (-4, -5). Answer: The magnitude of the vector \(\overrightarrow{P Q}\) is \(\sqrt{34}\).

Step 1: Identify the vector components

The components of the vector \(\overrightarrow{P Q}\) are the differences between the corresponding coordinates of points Q and P. In this case, we have: $$\Delta x = x_Q - x_P = -4 - (-7) = 3$$ $$\Delta y = y_Q - y_P = -5 - 0 = -5$$ So the vector \(\overrightarrow{P Q}\) can be represented as \((3, -5)\).

Step 2: Apply the Pythagorean theorem

To find the magnitude of the vector \(\overrightarrow{P Q}\), we apply the Pythagorean theorem on the components: $$\lVert\overrightarrow{P Q}\rVert = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}$$ The magnitude of the vector \(\overrightarrow{P Q}\) is \(\sqrt{34}\).

Key Concepts

Understanding Vector Components

When we talk about vector components, we're diving into how vectors can be broken down into smaller, manageable parts. Vectors represent both direction and magnitude, and in the context of coordinate geometry, they are described using their components along the x-axis and y-axis.

Consider a vector defined by two points, say point P and point Q, in a coordinate system. The vector from P to Q can be broken down into two components - one along the x-direction (\(\Delta x\)) and one along the y-direction (\(\Delta y\)).

• The x-component, \(\Delta x\), is derived by subtracting the x-coordinate of P from the x-coordinate of Q. It shows the "horizontal" change.
• The y-component, \(\Delta y\), represents the "vertical" change calculated by subtracting the y-coordinate of P from the y-coordinate of Q.

Breaking vectors into components makes it easier to perform calculations, such as finding their magnitude, using straightforward geometric principles.

Applying the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics that allows us to calculate distances in a plane. It's represented by the formula \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse of a right triangle. But how does this relate to vectors?

In vector analysis, the magnitude (or length) of a vector represented by components (\(\Delta x, \Delta y\)) resembles the hypotenuse of a right triangle formed by these components.

• The x-component and y-component act as the other two sides of the triangle.
• To find the magnitude, apply the Pythagorean theorem: \( \lVert\overrightarrow{P Q}\rVert = \sqrt{(\Delta x)^2 + (\Delta y)^2} \).

This method utilizes basic geometry to measure the "straight-line" distance a vector covers, conceptually treating components as orthogonal, right-angular segments of a path.

Delving into Coordinate Geometry

Coordinate geometry, sometimes referred to as analytical geometry, is a branch of mathematics that uses algebraic representation and analysis of geometrical shapes. This approach integrates geometry with algebra, allowing for precise description and calculation of geometrical figures using coordinates.

In the case of vectors in coordinate geometry, we assess them based on their positions in a Cartesian plane (which includes both x and y axes). Points in this plane are expressed as ordered pairs (x,y), which helps in establishing:

• Exact locations of points like P and Q.
• Defining vectors and their components with simple arithmetic.

By using this method, we make the abstract concept of vectors more tangible by linking them to specific points within a plane, facilitating operations like measuring distances or angles between said points through algebraic means.