Direction, distance & the dot product

Points and vectors

A point is a place: the red ball is at $p = (x, y, z)$, in metres from the robot's base. A vector is a displacement, an arrow with a length and a direction. Subtract two points and you get the vector from the first to the second:

$$v = p_{\text{ball}} - p_{\text{gripper}}$$

Length and direction

For $v = (v_x, v_y, v_z)$, the norm $\|v\| = \sqrt{v_x^2 + v_y^2 + v_z^2}$ is the vector's length: here, the distance from the gripper to the ball (np.linalg.norm(v)). Divide a vector by its norm and you get a unit vector $\hat v = v / \|v\|$ of length 1: a pure direction.

The dot product

The dot product turns two vectors $a$ and $b$ into one number:

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

$\theta$ is the angle between them. For unit vectors the dot product is $\cos\theta$: 1 when they point the same way, 0 when they are perpendicular, −1 when they are opposite. So $\theta = \arccos(\hat a \cdot \hat b)$. Rounding can push the cosine a hair past 1, and np.arccos then returns NaN, so clip it to $[-1, 1]$ first.

Projection

The dot product also splits a vector into parts. The part of $v$ along a direction $\hat u$ is its projection:

$$\operatorname{proj}_u v = (v \cdot \hat u)\,\hat u$$

$v \cdot \hat u$ says how far $v$ goes along $\hat u$ (in metres here), and multiplying by $\hat u$ turns that distance back into a vector. What's left, $v - \operatorname{proj}_u v$, is perpendicular to $\hat u$.

Jacobians, step 3 steers the gripper by commanding vectors like these as velocities.

Your task

1. Implement unit(v), angle_between(a, b) (in radians) and project(v, onto).
2. Seen from above, the arm reaches along the gripper's start position with $z$ set to 0. Find the ball whose position (also with $z = 0$) makes the smallest angle with that reach direction.
3. Move the gripper to the point 8 cm short of that ball on the straight line from where the gripper started to the ball's centre.

The starter draws an arrow from the gripper to every ball. Draw the chosen ball's arrow split into its part along the reach and the rest, then check that the two are perpendicular. The balls move on every run.