Dot Product & Projections

The dot product (also called scalar product) is one of the most important operations in robotics. It tells us how much two vectors point in the same direction and is essential for computing angles between robot links.

What is the Dot Product?

The dot product of two vectors produces a scalar (single number):

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

In 2D: $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y$

Geometric Interpretation

The dot product also equals:

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos(\theta)$

where $\theta$ is the angle between the vectors.

This formula connects:

• Algebraic definition (multiply components)
• Geometric meaning (angle between vectors)

The equivalence can be proven using the law of cosines. Place $\vec{a}$ and $\vec{b}$ tail-to-tail; the third side of the triangle is $\vec{a} - \vec{b}$. Expanding $|\vec{a} - \vec{b}|^2$ both algebraically (component-wise) and geometrically (law of cosines) yields the result.

Interactive Exploration

Adjust the vectors to see how the dot product changes with direction and magnitude:

Understanding the Result

The sign of the dot product tells us about the angle:

• Positive ($\vec{a} \cdot \vec{b} > 0$): Vectors point in similar directions ($\theta < 90°$)
• Zero ($\vec{a} \cdot \vec{b} = 0$): Vectors are perpendicular ($\theta = 90°$)
• Negative ($\vec{a} \cdot \vec{b} < 0$): Vectors point in opposite directions ($\theta > 90°$)

Computing Angles

We can rearrange the geometric formula to find angles:

$\theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)$

This is crucial in robotics for:

• Measuring joint angles
• Checking if the robot is at a desired configuration
• Collision detection
• Path planning

Vector Projection

The projection of $\vec{a}$ onto $\vec{b}$ is the “shadow” of $\vec{a}$ along $\vec{b}$‘s direction:

$\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$

The projection has magnitude:

$|\text{proj}_{\vec{b}}\vec{a}| = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{b}|} = |\vec{a}||\cos(\theta)|$

Properties of the Dot Product

1. Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
2. Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
3. Scalar multiplication: $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})$
4. Self dot product: $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Robotics Applications

1. Forward Kinematics Verification

Check if computed end-effector position matches expected direction.

2. Sensor Alignment

Determine if a sensor is pointing toward a target:

$\text{alignment} = \frac{\vec{sensor} \cdot \vec{to\_target}}{|\vec{sensor}||\vec{to\_target}|}$

If close to 1, sensor is well-aligned.

3. Field-of-View Check

Determine if a target point is within a sensor’s operational cone by comparing the direction to the target against the sensor’s boresight vector. If the dot product (normalized) exceeds a threshold, the target is within the field of view.

4. Obstacle Avoidance

Compute the component of velocity toward an obstacle:

$v_{toward} = \vec{v}_{robot} \cdot \hat{n}_{to\_obstacle}$

If positive, robot is approaching the obstacle.

Key Takeaways

• Dot product measures how aligned two vectors are
• Use it to compute angles: $\theta = \arccos\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\right)$
• Projection gives the component of one vector along another
• Critical for joint angle computation, force analysis, and navigation
• Perpendicular vectors have dot product zero

Practice Problems

1. Calculate $\vec{a} \cdot \vec{b}$ for $\vec{a} = (3, 4)$ and $\vec{b} = (2, 1)$.

2. Find the angle between $(1, 0)$ and $(1, 1)$.

3. Two robot links form vectors $(2, 1)$ and $(-1, 3)$. What’s the angle between them?

4. Project $\vec{a} = (4, 3)$ onto $\vec{b} = (1, 0)$. What’s the magnitude of the projection?

5. A force $(10, 5)$ N acts on a surface with normal $(0, 1)$. What’s the perpendicular component?