📚 Topic Summary
Orthogonal vectors are vectors that are perpendicular to each other. Mathematically, this means their dot product is zero. A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal. If, in addition, each vector in the orthogonal set has a length (or norm) of 1, then the set is called orthonormal. Understanding these concepts is crucial for many applications in linear algebra, including finding bases for vector spaces and solving systems of linear equations.
🧠 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Orthogonal Vectors |
a. A set of vectors where each vector has a norm of 1. |
| 2. Dot Product |
b. A set of vectors where every pair of distinct vectors is orthogonal. |
| 3. Orthogonal Set |
c. Vectors that are perpendicular to each other. |
| 4. Orthonormal Set |
d. The sum of the products of corresponding entries of two sequences of numbers. |
| 5. Vector Norm |
e. The length of a vector. |
Answer Key: 1-c, 2-d, 3-b, 4-a, 5-e
✏️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
Two vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^n$ are considered ________ if their ________ is equal to zero. An ________ set is a set of vectors, all of which are mutually orthogonal. If the vectors in an orthogonal set also have a ________ of 1, then the set is said to be ________.
Answer Key: orthogonal, dot product, orthogonal, norm, orthonormal
🤔 Part C: Critical Thinking
Explain, in your own words, why orthonormal bases are particularly useful in linear algebra. Give at least one example of where they are applied.
Example Answer: Orthonormal bases are useful because they simplify many computations. For example, finding the coordinates of a vector with respect to an orthonormal basis is straightforward, as it only requires computing dot products. Orthonormal bases are used extensively in Fourier analysis, signal processing, and quantum mechanics.