Vector Operations

Let's say we have two vectors, $v$ and $w$.

From the "list of numbers" perspective, the answer is incredibly simple. To add two vectors, you just add their corresponding components.

$v + w = [2+1, 1+3] = [3, 4]$

Geometrically, adding vectors is like combining a series of movements. Imagine a vector as a journey: "walk 2 steps east, then 1 step north."

To add vector $w$ to $v$, we simply start the journey of $w$ where the journey of $v$ ended.

1. Draw v: Start at the origin and draw the arrow for $[2, 1]$.
2. Draw w: Start at the tip of $v$ and draw the arrow for $[1, 3]$ (1 step east, 3 steps north).
3. The Result: The sum, $v + w$, is the new vector that starts at the origin and ends at the tip of the final vector, $w$.

This is the "tip-to-tail" rule, and it perfectly matches our numerical result! The final destination is indeed $[3, 4]$.

Let's take our vector $v = [2, 1]$ and multiply it by the scalar 3.

Just like with addition, the rule is simple: you multiply every component of the vector by the scalar.

$3 \times v = [3 \times 2, 3 \times 1] = [6, 3]$

Geometrically, multiplying a vector by a scalar scales its length.

• Multiplying by a scalar > 1 stretches the vector.
• Multiplying by a scalar between 0 and 1 shrinks it.
• Multiplying by a negative scalar flips its direction and then scales it.

The new vector $[6, 3]$ points in the exact same direction as $[2, 1]$, but it's exactly three times as long.