๐ Topic Summary
In linear algebra, understanding the angle between vectors and orthogonality is crucial. The angle between two vectors can be found using the dot product formula. Two vectors are orthogonal (perpendicular) if their dot product is zero. Mastering these concepts is fundamental for various applications, including physics and computer graphics.
๐ Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Vector |
A. A scalar representing the 'length' of a vector. |
| 2. Dot Product |
B. Two vectors are said to be this if their dot product equals zero. |
| 3. Magnitude |
C. A quantity with both magnitude and direction. |
| 4. Angle |
D. An operation that returns a scalar. |
| 5. Orthogonal |
E. The measure created by two intersecting lines. |
โ๏ธ Part B: Fill in the Blanks
Complete the following paragraph with the correct words.
Two vectors, $u$ and $v$, are _____ if their dot product, denoted as $u \cdot v$, is equal to _____. The _____ between two vectors can be found using the formula: $cos(\theta) = \frac{u \cdot v}{||u|| ||v||}$, where $||u||$ and $||v||$ represent the _____ of vectors $u$ and $v$, respectively. If the cosine of the angle is zero, then the vectors are _____.
๐ค Part C: Critical Thinking
Explain, in your own words, how the concept of orthogonality is useful in real-world applications. Provide at least one specific example.