Problem 13 - Algebra and Trigonometry

Question

Find the magnitude of the given vector. $$ \langle 3,5,-4\rangle $$

Step-by-Step Solution

Answer

The magnitude of the vector \( \langle 3,5,-4 \rangle \) is \( 5\sqrt{2} \).

Step 1: Understand the Formula for Magnitude

The magnitude of a vector \( \langle a, b, c \rangle \) is calculated using the formula \( \sqrt{a^2 + b^2 + c^2} \). This formula comes from the Pythagorean theorem in three dimensions.

Step 2: Identify Vector Components

Given the vector \( \langle 3, 5, -4 \rangle \), identify the components: \( a = 3 \), \( b = 5 \), and \( c = -4 \).

Step 3: Substitute Components into Formula

Substitute the components into the magnitude formula: \( \sqrt{3^2 + 5^2 + (-4)^2} \).

Step 4: Calculate the Squared Values

Calculate the squares of each component: \( 3^2 = 9 \), \( 5^2 = 25 \), \((-4)^2 = 16 \).

Step 5: Sum the Squared Values

Add the squares together: \( 9 + 25 + 16 = 50 \).

Step 6: Take the Square Root

Find the square root of the sum: \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that helps in calculating distances. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as: \[ c^2 = a^2 + b^2 \]
This theorem can be extended to three dimensions to help find the magnitude of vectors. Here, each vector component represents one of the triangle's sides. This is why we calculate the square root of the sum of the squared components, as seen in the magnitude formula:\[ \text{Magnitude} = \sqrt{a^2 + b^2 + c^2} \]
When you calculate the magnitude using this formula, you are essentially finding the length of the vector, just like a hypotenuse of a right-angled triangle in three-dimensional space.

3-dimensional vectors

Vectors in three-dimensional space have three components, each representing a direction along one of the spatial dimensions. For example, a vector \( \langle a, b, c \rangle \) is described by:

• \(a\): The component along the x-axis
• \(b\): The component along the y-axis
• \(c\): The component along the z-axis

These vectors help in describing diverse physical quantities like force, velocity, or any entity that has both magnitude and direction in 3D space.
To analyze these vectors, it's crucial to understand how to switch between its components and overall properties, such as its magnitude and direction. When dealing with problems in three dimensions, always ensure you keep track of which component corresponds to each axis, as mixing them up can lead to incorrect results.

vector components

To understand vectors thoroughly, one must first break them down into their components. For the vector \( \langle 3, 5, -4 \rangle \), the components are:

• \(3\): x-component
• \(5\): y-component
• \(-4\): z-component

These components tell you how much force or movement is in each direction. Essentially, they are like building blocks. When you sum them using the magnitude formula, you get a complete picture of the vector's length or 'how much' of the vector you have in 3D space.
Understanding vector components is crucial for performing operations like addition, subtraction, and dot product where each operation uses these components individually or in combination to produce a mathematical result that reflects real-world scenarios.