Problem 26 - Precalculus with Limits

Question

Finding the Magnitude of a Vector In Exercises \(25-30\) , use the dot product to find the magnitude of u. $$\mathbf{u}=\langle 4,-6\rangle$$

Step-by-Step Solution

Answer

The magnitude of the vector u is \( \sqrt{52} \).

Step 1 Identify the Components

The given vector is \(\mathbf{u}=\langle 4,-6\rangle\). So, the components of the vector u are 4 and -6.

Step 2 Apply the Magnitude Formula

The formula for the magnitude of a vector u with components a and b is \(\sqrt{a^2 + b^2}\). Thus substitute the components of vector u, which are 4 and -6, into the formula.

Step 3 Simplify

Upon substitution, the equation will look like this: \(\sqrt{4^2 + (-6)^2}\). Simplify that: \(\sqrt{16+36}= \sqrt{52}\).

Key Concepts

Dot Product

The dot product is a way to multiply two vectors together and results in a scalar. It's particularly useful for finding the angle between two vectors or determining orthogonality. Unlike regular multiplication, the dot product involves both the magnitude of the vectors and the cosine of the angle between them.
To calculate the dot product of two vectors, we use the formula:

• If vectors extbf{u} = a, b angle and extbf{v} = x, y angle, then the dot product is: extbf{u} cdot extbf{v} = ax + by.

This results in a single scalar value. For instance, with our given problem's vector extbf{u} = x, -6 angle, if we imagine a second vector extbf{v} = 1, 0 angle, and calculate the dot product, the result reveals important properties such as projections or angles.

Vector Components

Understanding vector components is essential for manipulating and analyzing vectors. Each vector can be broken down into its respective components, which represent the vector's influence in each dimension of its space.
For our vector extbf{u} = x, -6 angle, the components are:

• Horizontal component (x-axis): 4
• Vertical component (y-axis): -6

These components can be visualized on a graph as the two arrows originating from the origin, creating a right triangle. The hypotenuse of this triangle represents the vector extbf{u}. Analyzing the components allows us to assess how much the vector "moves" in each direction. This breakdown is fundamental for applications in physics, engineering, and computer graphics.

Vector Magnitude Formula

The vector magnitude formula is a key tool for measuring the length, or size, of a vector, often referred to as its 'magnitude'. To find it, use the Pythagorean theorem on the vector's components.
Given a vector extbf{u} = a, b . , the formula for its magnitude is:

• | extbf{u} | = a^2 + b^2 .

This formula stems from treating the vector's components as sides of a right triangle on a Cartesian plane. By substituting the components of extbf{u} = 6 , you plug them into the formula and compute: √(4^2 + (-6)^2).
After calculating, this gives √52. This formula is especially critical when comparing vector sizes or resolving forces in physics.