Problem 7 - Advanced Engineering Mathematics

Question

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{a} \cdot \mathbf{a}\)

Step-by-Step Solution

Answer

The dot product \(\mathbf{a} \cdot \mathbf{a}\) is 29.

Step 1: Understand the Dot Product

The dot product of two vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\) is given by the formula \(\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3\). When the dot product is calculated with the same vector (i.e., \(\mathbf{a} \cdot \mathbf{a}\)), it gives the square of the magnitude of that vector.

Step 2: Apply the Dot Product Formula

To find \(\mathbf{a} \cdot \mathbf{a}\), replace \(\mathbf{u}\) and \(\mathbf{v}\) with \(\mathbf{a} = \langle 2, -3, 4 \rangle\). So the calculation is: \(2 \times 2 + (-3) \times (-3) + 4 \times 4\).

Step 3: Calculate Each Term

Calculate each product separately: \(2 \times 2 = 4\), \((-3) \times (-3) = 9\), and \(4 \times 4 = 16\).

Step 4: Sum the Products

Add the results from each term to get the final dot product: \(4 + 9 + 16 = 29\).

Key Concepts

Understanding the Dot Product

The dot product is a fundamental operation when working with vectors in calculus. It combines two vectors to produce a scalar (a single number). Imagine you have two vectors, say \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \). The dot product is calculated using the formula:

• \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \)

When you perform a dot product with the same vector, like \( \mathbf{a} \cdot \mathbf{a} \), you get the square of the magnitude of that vector. Thus, it's not only about multiplying numbers; it's about finding relationships between the directions and magnitudes of two vectors.
The significance of the dot product lies in its wide application in physics and engineering, as it can tell us how much one vector goes in the direction of another.

Magnitude of a Vector Explored

The magnitude of a vector refers to its length or size. Imagine a vector as an arrow in space—a vector's magnitude is simply how long this arrow is. Mathematically, for a vector \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \), the magnitude \( ||\mathbf{a}|| \) is calculated using the formula:

• \( ||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2 + a_3^2} \)

This calculation involves squaring each of the vector's components, summing these squares, and then taking the square root of this sum. When you calculate the dot product of a vector with itself, \( \mathbf{a} \cdot \mathbf{a} \), you essentially find \( ||\mathbf{a}||^2 \).
Understanding the magnitude is crucial in many fields such as physics, where it can represent quantities like force or velocity. It gives us a way to understand the size of a vector without considering its direction.

Grasping Vector Operations

Vector operations, like the dot product, allow us to manipulate vectors and extract meaningful information from them. A few primary operations include:

• Addition: Combine corresponding components from each vector, like \( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \).
• Subtraction: Similar to addition, but you subtract corresponding components, like \( \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle \).
• Scalar Multiplication: Multiply each component by a scalar (single number), such as \( k \mathbf{a} = \langle ka_1, ka_2, ka_3 \rangle \).

Each of these operations helps us perform tasks like finding directions, calculating work, or analyzing forces in physics. They form the backbone of more complex analyses and problem-solving in mathematics and applied sciences.
Mastering vector operations lays the groundwork for effectively using vectors to model real-world phenomena.