Vectors

1. The dot product is the product of the magnitude of the vectors times the cosine of angle between them.

$$\vec a \cdot \vec b = |\vec a||\vec b|\cos\theta $$

2. The dot product of two vectors is 0 if and only if they are perpendicular. This holds for any dimensions.

$$\vec a \cdot \vec b = 0 \iff \vec a \perp \vec b$$

3. The dot product is the magnitude of a vector times the projection of the other vector on it.

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

4. The determinant of a 3 by 3 matrix is calculated as follows.
$$\det
\begin{pmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{pmatrix}$$$$
=a\det
\begin{pmatrix}
e&f\\
h&i
\end{pmatrix}-b\det
\begin{pmatrix}
d&f\\
g&i
\end{pmatrix}+c\det
\begin{pmatrix}
d&e\\
g&h
\end{pmatrix}$$
5. The cross-product is defined as

$$\vec a\times \vec b=\det
\begin{pmatrix}
\vec i&\vec j&\vec k\\
a_1&a_2&a_3\\
b_1&b_2&b_3
\end{pmatrix}$$

6. The magnitude of the cross-product is the product of the magnitude of the vectors times the sine of the angle between them.

$$|\vec a \times \vec b| = |\vec a||\vec b|\sin\theta$$

7. The volume of a parallelepiped formed by three vectors is:

$$V=\vec a\cdot (\vec b\times \vec c)=\det
\begin{pmatrix}
a_1&a_2&a_3\\
b_1&b_2&b_3\\
c_1&c_2&c_3
\end{pmatrix}$$

8. There are three ways to write a line in 3D space.

The vector form:

$$\vec r=\vec r_0+\vec v t$$

The parametric form:

$$\begin{cases}
x=x_0+at\\
y=y_0+bt\\
z=z_0+ct
\end{cases}$$

The symmetric form:

$$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$$

9. We need a normal vector and a point to specify a plane. If the normal vectors is \(\vec n=(a,b,c)\), and the point is \((x_0,y_0,z_0)\)

The equation of the plane is 

$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$

$$or$$

$$ax+by+cz+d=0$$

10.The distance from a point \(P(x_0,y_0,z_0)\) to a plane \(ax+by+cz+d=0\) is

$$D=\frac{|ax_0+by_0+cz_0+d|}{\sqrt{^2+b^2+c^2}}$$
11. The length of a curve is $$L=\int^{t_2}_{t_1}|\vec r'(t)|dt$$
12. The curvature of a curve is
$$\kappa= \frac{|\vec r'(t)\times \vec r”(t)|}{|\vec r'(t)|^3}$$