Problem 27 - Trigonometry

Question

Find the magnitude of each of the following vectors. $$\langle-5,6\rangle$$

Step-by-Step Solution

Answer

The magnitude of the vector \(\langle -5, 6 \rangle\) is approximately 7.81.

Step 1: Understand the Problem

To find the magnitude of a vector, we will use the formula for the magnitude of a 2D vector given by \(\langle a, b \rangle\), which is \(\sqrt{a^2 + b^2}\). In this problem, we need to find the magnitude of the vector \(\langle -5, 6 \rangle\).

Step 2: Identify the Components

The vector \(\langle -5, 6 \rangle\) has two components: \(a = -5\) and \(b = 6\). We will use these values in the magnitude formula.

Step 3: Plug into Magnitude Formula

Use the formula \(\sqrt{a^2 + b^2}\) with \(a = -5\) and \(b = 6\). This gives us \(\sqrt{(-5)^2 + 6^2}\).

Step 4: Calculate the Squares

Calculate \((-5)^2\) which equals 25, and \(6^2\) which equals 36.

Step 5: Sum the Squares

Add the squared values: \(25 + 36 = 61\).

Step 6: Compute the Square Root

Take the square root of the sum: \(\sqrt{61}\), which is approximately 7.81.

Key Concepts

Understanding 2D Vectors

When studying two-dimensional (2D) vectors, you are dealing with entities that have both direction and magnitude. Unlike regular numbers, these vectors are not values on a single axis but instead span across a plane. A 2D vector is typically represented by ordered pairs, such as \( \langle a, b \rangle \), where \( a \) is the horizontal component, and \( b \) is the vertical component. These components represent the direction and magnitude of the vector along the X and Y axes, respectively.

In practical terms, think of 2D vectors as arrows on a piece of paper. The tail of the arrow is at the origin (where x=0, y=0), and the head at the coordinate specified by the vector components. The direction of the arrow depends on the signs and values of these components, while the length of the arrow represents the vector's magnitude.

The Magnitude Formula Explained

The magnitude of a vector is a measure of its length, akin to how long a piece of string would be. For any 2D vector represented as \( \langle a, b \rangle \), you can calculate its magnitude using the formula: \( \sqrt{a^2 + b^2} \). This might seem similar to the Pythagorean Theorem, and that's because it works in the same principle.

The rationale behind this formula is simple: imagine a right triangle formed by the vector. The vector itself is the hypotenuse, and the legs of the triangle are the vector components \( a \) and \( b \). By squaring these components, adding the results, and taking the square root, you find the length of the vector. This provides the straight-line distance from the origin to the vector point, regardless of the vector's direction.

Digging Deep into Vector Components

Vector components are the building blocks of vectors, dictating both the direction and magnitude. In the vector \( \langle -5, 6 \rangle \), the components are \( a = -5 \) and \( b = 6 \).

These components tell us:

• The vector moves 5 units in the negative x-direction, meaning it extends to the left when visualized on a graph.
• It also rises 6 units in the positive y-direction, indicating an upward movement.

Together, they locate the precise end point of the vector starting from the origin.

Virtually every problem involving vectors will require you to understand these two components, as they define how the vector behaves in a coordinate system. By analyzing these components, you can gain insight into the vector's overall influence in a space, including its path and positioning.