📚 Topic Summary
In mathematics, particularly linear algebra, $R^n$ represents the $n$-dimensional real coordinate space. Vector operations in $R^n$ involve manipulating vectors, which are ordered lists of $n$ real numbers. These operations include vector addition, scalar multiplication, dot product, and cross product (specifically in $R^3$). Understanding these operations is fundamental for solving problems in various fields like physics, engineering, and computer graphics. This worksheet provides practice problems to reinforce your understanding of these concepts.
🧮 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Vector Addition |
A. A real number that scales a vector. |
| 2. Scalar Multiplication |
B. An ordered list of $n$ real numbers. |
| 3. Dot Product |
C. The sum of two vectors, component-wise. |
| 4. Vector |
D. An operation that results in a scalar, calculated by summing the products of corresponding components. |
| 5. Magnitude |
E. The length of a vector, calculated as the square root of the sum of the squares of its components. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
The ________ of a vector represents its length. ________ involves multiplying a vector by a scalar, changing its magnitude but not its direction (unless the scalar is negative). ________ is performed by adding corresponding components of two vectors. The ________ of two vectors results in a scalar value.
🤔 Part C: Critical Thinking
Explain how vector operations in $R^n$ are used in computer graphics to manipulate objects in 3D space. Provide a specific example.