Problem 12 - Precalculus Mathematics for Calculus

Question

Find the magnitude of the given vector. $$\langle 5,0,-12\rangle$$

Step-by-Step Solution

Answer

The magnitude is 13.

Step 1: Identify the Vector Components

The given vector is \( \langle 5, 0, -12 \rangle \). It has three components: 5, 0, and -12 in the x, y, and z directions, respectively.

Step 2: Use the Formula for Magnitude of a Vector

The magnitude of a vector \( \langle a, b, c \rangle \) is given by the formula \( \sqrt{a^2 + b^2 + c^2} \). Here, \( a = 5 \), \( b = 0 \), and \( c = -12 \).

Step 3: Substitute the Components into the Formula

Substitute the components into the magnitude formula: \( \sqrt{5^2 + 0^2 + (-12)^2} \).

Step 4: Calculate Each Square

Calculate each term inside the square root separately: \( 5^2 = 25 \), \( 0^2 = 0 \), \( (-12)^2 = 144 \).

Step 5: Sum the Squares

Add the squared components: \( 25 + 0 + 144 = 169 \).

Step 6: Find the Square Root

Take the square root of the sum: \( \sqrt{169} = 13 \).

Step 7: State the Magnitude of the Vector

The magnitude of the vector \( \langle 5, 0, -12 \rangle \) is 13.

Key Concepts

3D Vectors

A 3D vector is a mathematical entity that has both direction and magnitude in three-dimensional space. Think of a 3D vector as an arrow pointing from one point in space to another. It is often represented by three components, corresponding to the x, y, and z axes. For example, the vector \( \langle 5, 0, -12 \rangle \) tells us that the arrow points 5 units in the x-direction, 0 units in the y-direction, and -12 units in the z-direction.
Understanding 3D vectors is essential in physics and engineering, where they are used to represent quantities like force, velocity, and displacement. Recognizing that these vectors provide a comprehensive way to note both direction and distance in space is key to visualizing their applications.

Vector Components

Vector components are the building blocks of a vector. Each component represents the vector's influence along one of the coordinate axes. For a 3D vector, these axes are x, y, and z. The vector \( \langle 5, 0, -12 \rangle \) has three components:

• 5 along the x-axis (rightward direction)
• 0 along the y-axis (no movement up or down)
• -12 along the z-axis (downward direction)

These components act as coordinate points that together define the vector's direction and length in 3D space. By analyzing each component, one can fully understand how the vector affects movements and positions in each direction.
Knowing how to work with vector components helps in dissecting and understanding vector operations, such as addition and scalar multiplication.

Magnitude Calculation

The magnitude of a vector is essentially its length. To calculate the magnitude of a 3D vector, you use the Pythagorean Theorem extended into three dimensions. This involves squaring each component of the vector, summing them up, and then taking the square root of the result.
For our vector \( \langle 5, 0, -12 \rangle \), you'll plug the components into the formula for magnitude: \[ \sqrt{a^2 + b^2 + c^2} \]Substitute the components: \[ \sqrt{5^2 + 0^2 + (-12)^2} \]This method ensures a precise calculation that reflects the true length of the vector in space, regardless of orientation.

Square Root Calculation

Square root calculation is the final step in finding the magnitude of a vector. After summing the squares of each component, the square root gives us the actual magnitude. For instance, with our vector, after calculating and summing the squares as \( 5^2 = 25 \), \( 0^2 = 0 \), and \( (-12)^2 = 144 \), the total is 169.
To find the magnitude, calculate \( \sqrt{169} \), which results in 13. This value represents the vector's length in 3D space. Recognizing the purpose of square roots in such calculations helps in understanding how complex multimodal measurements (like vectors) boil down to a single meaningful length.

• Discovering that even if components have negative values, their squares remain positive in calculations.
• Using clear and precise computational steps ensures accuracy and clear communication of concepts.