Notes

a·b = |a||b|cos(theta). Measures how aligned two vectors are, not distance, not area.

Math

$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z$$

Geometric meaning: $\vec{a} \cdot \vec{b}$ is $|\vec{a}|$ times the signed length of $\vec{b}$‘s projection onto $\vec{a}$‘s direction (or symmetrically the other way). If $\vec{a}$ is a unit vector, $\vec{a} \cdot \vec{b}$ IS that signed projection length directly.

What the result means:

• Sign only, direction-agnostic magnitude aside: $\vec{a} \cdot \vec{b} > 0 \Rightarrow \theta < 90°$, $= 0 \Rightarrow$ perpendicular, $< 0 \Rightarrow \theta > 90°$
• Full angle if you need it (expensive, avoid unless necessary): $\theta = \arccos\left(\frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\right)$
• If both vectors are unit length, the dot product is bounded to $[-1, 1]$ and directly equals $\cos\theta$

Code

float dot = glm::dot(a, b);
float projLength = glm::dot(a, glm::normalize(b));  // scalar projection of a onto b

Uses in graphics

• Back-face culling — sign of N·V
• Diffuse (Lambertian) lighting: intensity ∝ max(0, N·L)
• Specular highlights (Blinn-Phong): N·H term
• Which side of a plane a point is on: sign of planeNormal · (point - planePoint)
• Camera frustum / visibility checks: dot(cameraForward, dirToObject) to reject anything behind the camera
• Closest point on a line/ray to another point (scalar projection)

Gotchas

• Dot product alone can’t tell you the exact angle, only which side of 90° you’re on — use the sign, don’t reach for acos unless you actually need the number.
• Only equals cos(theta) directly when both vectors are normalized; otherwise it’s scaled by both magnitudes, don’t forget to divide if you need the pure angle relationship.

Related

• Cross product
• Back-face culling