Scalars

• Scalars $\alpha, \beta, \gamma$ from a scalar field
• Operations $\alpha + \beta$, $\alpha \cdot \beta$ , $0$, $1$, $-\alpha$, $()^{-1}$

Vectors

• Vectors $u, v, w$ from a vector space
• Vector addition $u + v$, subtraction $u - v$
• Zero vector $0$
• Scalar multiplication $\alpha v$

Lines and line Segments

• Parametric form of line: $P(\alpha) = P_0 + \alpha d$
• Line segment between $Q$ and $R$: $P(\alpha) = (1 - \alpha) Q + \alpha R$ $for \; 0 \leq \alpha \leq 1$

Dot Product (Projection)

• Dot Product projects one vector onto another vector
• $u \cdot v=u_1 v_1+ u_2 v_2 + u_3 v_3 = |u||v|\cos(\theta)$
• $pr_vu = (u \cdot v)v/{|v|}^2$

Cross Product

• $|a \times b| = |a||b||\sin(\theta)|$
  $\begin{pmatrix}a_1 \\ a_2 \\a_3\end{pmatrix} \times \begin{pmatrix}b_1 \\ b_2 \\ b_3\end{pmatrix} = \begin{pmatrix}a_2b_3 - a_3b_2 \\ a_3b_1 - a_1b_3 \\ a_1b_2 - a_2b_1\end{pmatrix}$
• Cross product is perpendicular to both $a$ and $b$

• Right- hand rule

Planes

• A Plane can be defined by a point and two vectors or by three points
• $P(\alpha, \beta) = R + \alpha u + \beta v$

• $P(\alpha, \beta) = R + \alpha(Q - R) + \beta(P - Q)$

Planes and normal

• Plane defined by Point $P_0$ and vectors $u$ and $v$
• $u$ and $v$ should not be parallel
• Parametric form: $T(\alpha, \beta) = P_0 + \alpha u + \beta v$ ($\alpha$ and $\beta$ are scalars)
• $n = u \times v / |u \times v|$ is the normal
• $n \cdot (P - P_0) = 0$ if and only if $P$ lies in plane