Vector Operations

In this lesson you’ll learn the basic operations you can perform on vectors: addition, subtraction, scalar multiplication, and the dot product.

Vectors are added component-wise:

$$\mathbf{u} + \mathbf{v} = \langle u_1 + v_1,\; u_2 + v_2,\; u_3 + v_3 \rangle$$

Subtraction works the same way:

$$\mathbf{u} - \mathbf{v} = \langle u_1 - v_1,\; u_2 - v_2,\; u_3 - v_3 \rangle$$

Geometrically, vector addition follows the triangle rule (or parallelogram rule). Place the tail of $\mathbf{v}$ at the tip of $\mathbf{u}$, and the result $\mathbf{u} + \mathbf{v}$ goes from the origin to the tip of $\mathbf{v}$:

In the diagram above: the blue arrow is $\mathbf{u} = \langle 2, 3 \rangle$ starting from the origin. The green arrow is $\mathbf{v} = \langle 5, -1 \rangle$ placed at the tip of $\mathbf{u}$. The dashed orange arrow shows the result $\mathbf{u} + \mathbf{v} = \langle 2+5,\; 3+(-1) \rangle = \langle 7, 2 \rangle$, going directly from the origin to where $\mathbf{v}$ ends. This is the triangle rule in action.

Multiplying a vector by a number (scalar) scales its length:

$$c \cdot \mathbf{v} = \langle c\, v_1,\; c\, v_2,\; c\, v_3 \rangle$$

• If $\lvert c \rvert \gt 1$, the vector gets longer
• If $0 \lt \lvert c \rvert \lt 1$, the vector gets shorter
• If $c$ is negative, the vector reverses direction

The diagram shows three versions of the same direction. The blue arrow is $\mathbf{v} = \langle 2, 1 \rangle$. The green arrow is $2\mathbf{v} = \langle 4, 2 \rangle$, which points the same way but is twice as long. The orange arrow is $-\mathbf{v} = \langle -2, -1 \rangle$, which has the same length as $\mathbf{v}$ but points in the opposite direction. Multiplying by a positive scalar stretches; multiplying by a negative scalar flips.

The dot product (also called scalar product) of two vectors returns a single number:

$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$$

It can also be written as:

$$\mathbf{u} \cdot \mathbf{v} = \lVert \mathbf{u} \rVert \, \lVert \mathbf{v} \rVert \cos \theta$$

where $\theta$ is the angle between them.

Important properties:

• If $\mathbf{u} \cdot \mathbf{v} = 0$, the vectors are orthogonal (perpendicular).
• The dot product tells you how much one vector points in the direction of another.

Here $\mathbf{u} = \langle 3, 4 \rangle$ (blue) and $\mathbf{v} = \langle -2, 5 \rangle$ (green) meet at the origin with a 59-degree angle between them (purple arc). The dot product $\mathbf{u} \cdot \mathbf{v} = (3)(-2) + (4)(5) = 14$. Since the result is positive, the vectors point in roughly the same general direction (the angle is less than 90 degrees). If the dot product were zero, they’d be perpendicular; if negative, they’d point away from each other.

Example 1: Addition and Subtraction

Let $\mathbf{u} = \langle 2, 3 \rangle$ and $\mathbf{v} = \langle 5, -1 \rangle$

$\mathbf{u} + \mathbf{v} = \langle 7, 2 \rangle$

$\mathbf{u} - \mathbf{v} = \langle -3, 4 \rangle$

Example 2: Scalar Multiplication

Let $\mathbf{w} = \langle 1, 2, 3 \rangle$

$3\mathbf{w} = \langle 3, 6, 9 \rangle$

$-2\mathbf{w} = \langle -2, -4, -6 \rangle$ (reverses direction and doubles length)

Example 3: Dot Product

$\mathbf{u} = \langle 3, 4 \rangle$, $\mathbf{v} = \langle -2, 5 \rangle$

$\mathbf{u} \cdot \mathbf{v} = (3)(-2) + (4)(5) = -6 + 20 = 14$

Angle check: $\cos \theta = \frac{14}{5 \times \sqrt{29}} \approx 0.52$, so $\theta \approx 59°$

Vector operations are fundamental in game development:

• Movement: Add velocity vector to position vector each frame
• Forces: Add multiple force vectors (gravity, thrust, friction) to get net force
• Lighting: Dot product between surface normal and light direction determines brightness (Lambertian shading)
• AI: Dot product helps determine if an enemy is facing toward or away from the player

Example: When your character jumps, you add a gravity vector (pointing down) to the current velocity vector every frame.