Problem 30 - Trigonometry

Question

Find the magnitude of each of the following vectors. $$\langle-7,0\rangle$$

Step-by-Step Solution

Answer

The magnitude of the vector \( \langle -7, 0 \rangle \) is 7.

Step 1: Understand the Vector Components

The given vector is \( \langle -7, 0 \rangle \). It consists of two components: \( x = -7 \) and \( y = 0 \). To find its magnitude, we will use the formula for the magnitude of a vector in two-dimensional space.

Step 2: Apply the Magnitude Formula

The formula for calculating the magnitude of a vector \( \langle x, y \rangle \) is \( \sqrt{x^2 + y^2} \). We will substitute \( x = -7 \) and \( y = 0 \) into this formula.

Step 3: Calculate Each Component's Square

Calculate \( x^2 \, \rightarrow \, (-7)^2 = 49 \). Calculate \( y^2 \, \rightarrow \, 0^2 = 0 \).

Step 4: Add the Squares

Add the results from the previous step: \( 49 + 0 = 49 \).

Step 5: Find the Square Root

The magnitude is the square root of the sum from the previous step: \( \sqrt{49} = 7 \).

Key Concepts

Understanding Vector Components

Vectors are essential elements in physics and mathematics, as they depict both magnitude and direction. In two-dimensional space, a vector is often represented as \( \langle x, y \rangle \), where \( x \) and \( y \) are called the "components" of the vector. These components specify the vector's effect along the x-axis and y-axis respectively.
For example, in the vector \( \langle -7, 0 \rangle \), the component \( x = -7 \) explains how far along the x-axis the vector extends, while \( y = 0 \) shows no movement along the y-axis.

• X-component: This is \( -7 \), indicating a movement backwards or negative along the x-axis.
• Y-component: This is 0, suggesting no vertical change or influence.

Understanding the components helps in visualizing the direction and alignment of the vector in the coordinate plane.

Applying the Magnitude Formula

The magnitude of a vector is akin to its length or size, regardless of its direction. To compute the magnitude of a vector \( \langle x, y \rangle \), we use the magnitude formula:
\[\text{Magnitude} = \sqrt{x^2 + y^2}\]
Essentially, this formula is derived from the Pythagorean theorem, where the vector forms the hypotenuse of a right triangle formed by its components as legs. For the vector \( \langle -7, 0 \rangle \):

• Plug in the components: with \( x = -7 \) and \( y = 0 \), substitute them into the formula.
• The necessary operation is squaring each component: \((-7)^2 = 49\) and \(0^2 = 0\).

By understanding the magnitude formula, you equip yourself with a method to measure the "length" of a vector in any directional plane.

Executing Vector Calculations

After substituting into the magnitude formula \( \sqrt{x^2 + y^2} \), the next step is calculating smoothly by following these concise procedures:

• Square Each Component: Convert each component into its square to avoid considering directional signs at this stage.
• Add the Squares: Sum the individual squares: \(49 + 0 = 49\).
• Compute the Square Root: To extract the magnitude, take the square root of the summed squares: \(\sqrt{49} = 7\).

The resulting magnitude presents how long or large the vector truly is, illustrating the powerful blend of vector components and math's spatial measurements. Understanding these calculations provides not only a magnitude but also strengthens comprehension of vector operations in the broader mathematical context.