Problem 28 - Trigonometry

Question

Find the magnitude of each of the following vectors. $$\langle-9,-2\rangle$$

Step-by-Step Solution

Answer

The magnitude of the vector \( \langle -9, -2 \rangle \) is \( \sqrt{85} \).

Step 1: Understand the Formula

To find the magnitude of a vector \( \langle a, b \rangle \), we use the formula \( \sqrt{a^2 + b^2} \). This formula comes from the Pythagorean theorem, which relates the lengths of the sides of a right triangle.

Step 2: Substitute Values

Substitute the components of the vector \( \langle -9, -2 \rangle \) into the formula: \( a = -9 \) and \( b = -2 \). So the magnitude is \( \sqrt{(-9)^2 + (-2)^2} \).

Step 3: Calculate the Squares

Calculate the squares of the components: \((-9)^2 = 81\) and \((-2)^2 = 4\). Substitute these into the formula to get \( \sqrt{81 + 4} \).

Step 4: Add the Squares

Add the squares calculated in the previous step: \(81 + 4 = 85\). So the magnitude is now expressed as \( \sqrt{85} \).

Step 5: Simplify the Expression

Calculate the square root to find the magnitude: \( \sqrt{85} \). Since 85 cannot be simplified further into rational numbers, \( \sqrt{85} \) is the precise magnitude.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics that relates the sides of a right triangle. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as:

• \[ c^2 = a^2 + b^2 \]

where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.
In the context of vectors, the Pythagorean theorem helps us to find the magnitude of a vector. Imagine a vector as the hypotenuse of a right triangle, with its components being the triangle's other two sides. By applying the theorem, the magnitude of a vector \( \langle a, b \rangle \) can be calculated as follows:

• \[ \sqrt{a^2 + b^2} \]

This is essentially using the Pythagorean theorem to find the length or magnitude of the vector, which represents the hypotenuse in a triangle formed by the vector components.

Vector Components

A vector is a mathematical entity that has both magnitude and direction. Typically represented as \( \langle a, b \rangle \) in a Cartesian plane, the vector components \( a \) and \( b \) correspond to its influence along the horizontal and vertical axes, respectively.
Understanding vector components is crucial for visualizing how a vector behaves in space. Here’s how it works:

• The component \( a \) is how much the vector moves along the x-axis.
• The component \( b \) is how much the vector moves along the y-axis.

These components can be positive, negative, or even zero, depending on the vector's direction.
To find the magnitude of a vector, we consider these components as the sides of a right triangle. By applying the Pythagorean theorem, we calculate how far away the endpoint of the vector is from the origin. Thus, magnitude isn't just a geometric length; it tells us the overall pull or push the vector represents in its direction.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. It plays a crucial role in understanding vectors because it provides tools to determine direction and angle.When working with vectors, trigonometry helps us understand:

• How to split a vector into its components using trigonometric functions like sine and cosine.
• The angle a vector makes with an axis, often found using inverse trigonometric functions.

For example, if a vector has an angle \( \theta \) with the x-axis, its components can be expressed using trigonometric functions:

• The horizontal component (x-axis) is \( a = |\mathbf{v}| \cdot \cos(\theta) \).
• The vertical component (y-axis) is \( b = |\mathbf{v}| \cdot \sin(\theta) \).

By understanding these concepts, we can reconstruct the vector from its components or calculate its direction, enhancing our comprehension of how vectors function and interact in different contexts.