Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ \mathbf{a} \cdot \mathbf{a}=\|\mathbf{a}\|^{2} $$

Step-by-Step Solution

Answer

Yes, \( \mathbf{a} \cdot \mathbf{a} = \| \mathbf{a} \|^2 \) because both represent the sum of squares of the components of \( \mathbf{a} \).

Step 1: Understand the Dot Product

The dot product of a vector \( \mathbf{a} = (a_1, a_2, a_3) \) with itself is given by \( \mathbf{a} \cdot \mathbf{a} = a_1^2 + a_2^2 + a_3^2 \). Essentially, the dot product relates to the addition of the squares of each component of the vector.

Step 2: Relate the Dot Product to the Magnitude of a Vector

The magnitude (or length) of a vector \( \mathbf{a} \) is given by the formula \( \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2 + a_3^2} \). Therefore, when we square the magnitude, we get \( \| \mathbf{a} \|^2 = a_1^2 + a_2^2 + a_3^2 \).

Step 3: Equate Dot Product and Squared Magnitude Formula

By comparison, we see that \( \mathbf{a} \cdot \mathbf{a} = a_1^2 + a_2^2 + a_3^2 \) is exactly equal to \( \| \mathbf{a} \|^2 = a_1^2 + a_2^2 + a_3^2 \). Thus, we can conclude that the two expressions are equivalent.

Key Concepts

Vector Magnitude

The magnitude of a vector, often called its length, gives us an idea of how long the vector is in space. For a vector \( \mathbf{a} = (a_1, a_2, a_3) \), the magnitude is found using the formula:\[\| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]This formula is derived from the Pythagorean Theorem. Each component of the vector \((a_1, a_2, a_3)\) can be thought of as forming the sides of a right triangle in space.
When combined, they create a hypotenuse that represents the vector itself.
- The power of the magnitude is that it offers a single number that characterizes the size of the vector.- It's a measure of distance from the origin to the vector's endpoint in three-dimensional space.To make sense of magnitude, picture it as the length of a straight stick that stretches from one point to another without bending. With this intuitive grasp, understanding vectors becomes a lot easier.

Vector Components

Breaking down a vector into its components helps us understand its direction and effect in space. A vector component shows how much a vector points in any direction (usually along the x, y, and z axes).
When we look at a vector like \(\mathbf{a} = (a_1, a_2, a_3)\), each \(a_i\) is a component.- **X component**: Represents movement along the x-axis, which is \(a_1\).- **Y component**: Represents movement along the y-axis, which is \(a_2\).- **Z component**: Represents movement along the z-axis, which is \(a_3\).These components let us view the impact a vector has when acting in space. Each number tells you how far and in what direction you travel along the axes to "reach" the vector.
Think of components as the coordinates in a city map that guide you to a specific location.The components also allow us to rebuild the vector by saying it heads in this much in the x-direction, this much in the y-direction, and this much in the z-direction. Each component forms the complete vector when combined appropriately.

Properties of Vectors

Vectors have unique properties that help us understand how they behave and interact in mathematical and physical contexts.
Some important properties include:- **Commutative Property**: The order in which you add vectors doesn't matter. For example, \(\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}\).- **Associative Property**: Grouping of vector addition can be rearranged without changing the result, i.e., \((\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})\).- **Scalar Multiplication**: When you scale a vector by a number (scalar), you change its magnitude but not its direction unless the scalar is negative. If \(m\) is a scalar, then \(m\mathbf{a}\) stretches or shrinks \(\mathbf{a}\).- **Distributive Property**: Scalars distribute over vector addition. This means \(m(\mathbf{a} + \mathbf{b}) = m\mathbf{a} + m\mathbf{b}\).These properties enable vectors to be manipulated in calculations while maintaining their intrinsic attributes. Interacting vectors through properties like these ensures that the calculus and geometry involving vectors remain consistent and predictable. By mastering these properties, you gain flexibility and precision in handling vectors, as they form the foundational bedrock for more advanced topics in physics and engineering.