# Geometry

## Dot Product of Vectors

Dot product is also known as **scalar product** or **inner product**.

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.

$$
\bm{a}\cdot \bm{b} = \sum\_{i=1}^na\_ib\_i
$$

Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.

$$
\bm{a}\cdot \bm{b}=\lvert \bm{a}\rvert\lvert \bm{b}\rvert\cos\theta
$$

### Usage

1. Calculate the angle between two vectors.

$$$
\cos\theta =\frac{\bm{a}\cdot \bm{b}}{\lvert \bm{a}\rvert\lvert \bm{b}\rvert}$$ 2. If two vectors are perpendicular, the dot product is `0`. ### Problem \* [963. Minimum Area Rectangle II (Medium)](https://leetcode.com/problems/minimum-area-rectangle-ii/) ## Cross Product of Vectors Cross product is also known as **vector product** ((occasionally **directed area product**, to emphasize its geometric significance) ) The cross product $\bm{a}\times\bm{b}$ is defined as $\bm{a}$ vector $\bm{c}$ that is perpendicular (orthogonal) to both $\bm{a}$ and $\bm{b}$, with $\bm{a}$ direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
$$$

\bm{a}\times\bm{b}=\lvert\bm{a}\rvert\lvert\bm{b}\rvert\sin(\theta)\bm{n}

$$
where $\bm{n}$ is a unit vector perpendicular to the plane containing $\bm{a}$ and $\bm{b}$. ![Finding the direction of the cross product by the right-hand rule.](../.gitbook/assets/cross-product-right-hand-rule.png) ## Check if three points are one te same line Say we have 3 points, `(x1, y1)`, `(x2, y2)` and `(x3, y3)`. Check if they are on the same line. Normally we can check if slopes are the same.
$$

\frac{y\_2-y\_1}{x\_2-x\_1} = \frac{y\_3-y\_2}{x\_3-x\_2}

$$
To avoid: 1. The precision error we might get from the divisions 2. Divide by zero issue We can use the following equation:
$$

(y\_2-y\_1)\cdot(x\_3-x\_2) = (y\_3-y\_2)\cdot(x\_2-x\_1)

$$
Example question: [2280. Minimum Lines to Represent a Line Chart (Medium)](https://leetcode.com/problems/minimum-lines-to-represent-a-line-chart) ## Get angle between two points Assume we have two points `(x1, y1)` and `(x2, y2)`, we want to get the angle between these two points `cpp if ((x1 > x2) || (x1 == x2 && y1 < y2)) swap(x1, x2), swap(y1, y2); // make sure these two points are ordered atan2(x2 - x1, y2 - y1) ` Example question: [149. Max Points on a Line (Hard)](https://leetcode.com/problems/max-points-on-a-line)
$$