Lesson 2: Vector Operations

1. Vector Addition

To add two vectors, add their corresponding components.

Geometric meaning: place tail of second vector at head of first → result is from start to end.

Example:

\[ \begin{pmatrix} 2 \\ 1 \end{pmatrix} + \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix} \]

2. Vector Subtraction

Subtract corresponding components.

Geometric meaning: reverse the second vector and add.

Example:

\[ \begin{pmatrix} 5 \\ 7 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \]

3. Scalar Multiplication

Multiply every component by a number (scalar).

Effect: stretches or shrinks the vector; negative scalar reverses direction.

Example:

\[ 3 \times \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 6 \end{pmatrix}, \quad -1 \times \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} -4 \\ 1 \end{pmatrix} \]

Summary – Lesson 2

• Add / subtract → component-wise
• Scalar multiply → multiply every component
• These operations preserve the linear structure

Next: How long is a vector? (magnitude)