14.1: Dot Product
- Page ID
- 14858
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The dot product (or scalar product) is the sum of the products of the individual entries of two vectors. Let hata = (a1, . . . , an) and ˆb= (b1, . . . , bn) be two vectors. Then, their dot product â · ˆb is given by
\[\hat{a}\cdot \hat{b}=\sum_{i}^{n}a_{i}b_{i}\]
The dot product therefore takes two sequences of numbers and returns a single scalar. In robotics, the dot product is mostly relevant due to its geometric interpretation:
\[\hat{a}\cdot \hat{b}=\left \| \hat{a} \right \|\left \| \hat{b} \right \|\cos\theta \]
with θ the angle between vectors â and ˆb
If â and ˆb are orthogonal, it follows â · ˆb = 0. If â and ˆb are parallel, it follows â · ˆb = ||â|| ||ˆb||.