Dot product.

Definition 1.6   Let $\overrightarrow{u}$and $\overrightarrow{v}$be two vectors in $\mathcal{E}$. The dot product, or scalar product of $\overrightarrow{u}$and $\overrightarrow{v}$is the real number

$$\overrightarrow{u}\cdot \overrightarrow{v} = \vert \overright... ...row{v} \vert \cos ( \overrightarrow{u} , \overrightarrow{v} )$$

Proposition 1.7       

1.

$\forall \overrightarrow{u}$, $\overrightarrow{u}\cdot \overrightarrow{u}\geq 0$.

2.

$\overrightarrow{u}\cdot \overrightarrow{u} = 0 \Longleftrightarrow \overrightarrow{u} = \overrightarrow{0}$.

3.

$\forall \overrightarrow{u} ,\overrightarrow{v}$, $\overrightarrow{u}\cdot \overrightarrow{v} = \overrightarrow{v}\cdot \overrightarrow{u} .$

4.

$\forall \overrightarrow{u} ,\overrightarrow{v} ,\overrightarrow{w}$, $\overrightarrow{u}\cdot (\overrightarrow{v} + \overrightarrow{w} ) = \overrightarrow{u}\cdot \overrightarrow{v} + \overrightarrow{u}\cdot \overrightarrow{w}$.

5.

$\forall \overrightarrow{u} ,\overrightarrow{v} ,\overrightarrow{w}$, $( \overrightarrow{u} + \overrightarrow{v} ) \cdot \overrightarrow{w} ) = \over... ...tarrow{u}\cdot \overrightarrow{w} + \overrightarrow{v}\cdot \overrightarrow{w}$.

6.

$\forall \overrightarrow{u} ,\overrightarrow{v}$, $\forall \lambda \in \mathbb{R}$, $\overrightarrow{u}\cdot \lambda \overrightarrow{v} = \lambda \overrightarrow{u}\cdot \overrightarrow{v} = \lambda (\overrightarrow{u}\cdot \overrightarrow{v} ) .$

Proposition 1.8   Two vectors $\overrightarrow{u}$and $\overrightarrow{v}$are orthogonal if, and only if, $\overrightarrow{u}\cdot \overrightarrow{v} = 0$.

Example 1.9   If $\overrightarrow{u} = \begin{pmatrix}1\\ 2\\ 3 \end{pmatrix}$and $\overrightarrow{v} = \begin{pmatrix}1 \\ 1\\ -1 \end{pmatrix}$, then $\overrightarrow{u}\cdot \overrightarrow{v} = 1 \cdot 1 + 1 \cdot 2 + (-1) \cdot 3 = 0$. Thus $\overrightarrow{u}\perp \overrightarrow{v}$.

Definition 1.10   Let $\mathcal{P}$be a plane whose cartesian equation is ax+by+cz+d=0. The vector $\overrightarrow{N} = \begin{pmatrix}a \\ b \\ c \end{pmatrix}$is called a normal vector for $\mathcal{P}$.

 

  

Figure 5: A vector normal to a plane.

 

Example 1.11   The vector $\overrightarrow{N} =\begin{pmatrix}1 \\ 2 \\ -3 \end{pmatrix}$is normal to the plane $\mathcal{P}$whose equation is x + 2y -3z +5 =0.

Proposition 1.12   Let $\mathcal{P_1}$and $\mathcal{P_2}$be two planes with respective normal vectors $\overrightarrow{N} _1$and $\overrightarrow{N} _2$. The two planes $\mathcal{P_1}$and $\mathcal{P_2}$are orthogonal if, and only if, the vectors $\overrightarrow{N} _1$and $\overrightarrow{N} _2$are orthogonal.

Example 1.13   Let $\mathcal{P}_1: \; 2x-y+3z-1=0$and $\mathcal{P}_2: \; 4x+2y-2z+5=0$. These planes have respective normal vectors $\overrightarrow{N} _1= \begin{pmatrix}2 \\ -1 \\ 3 \end{pmatrix}$and $\overrightarrow{N} _2= \begin{pmatrix}4 \\ 2 \\ -2 \end{pmatrix}$. We have: $\overrightarrow{N} _1 \cdot \overrightarrow{N} _2 = 2 \cdot 4 + (-1) \cdot 2 + 3 \cdot (-2) = 0$, thus $\mathcal{P}_1 \perp \mathcal{P}_2$.

Proposition 1.14   A line $\mathcal{L}$and a plane $\mathcal{P}$are perpendicular if, and only if, a direction vector for $\mathcal{L}$is a normal vector for $\mathcal{P}$.

Example 1.15   Let $\mathcal{L}$be the line given by the parametric equations

$$\begin{cases} x=1+2t \\ y=-5+4t \\ z=3-3t \end{cases}\; , \; t \in \mathbb{R} .$$

and let $\mathcal{P}$be the plane whose cartesian equation is 2x+4y-3z-1=0. Then $\mathcal{L}$and $\mathcal{P}$are perpendicular.

Example 1.16   If $\mathcal{P}$is defined by the equation 3x-y+5z-6=0, then the line $\mathcal{L}$through the point A(5,-2,7) and perpendicular to $\mathcal{P}$has the following parametric equations:

$$\begin{cases} x=5+3t \\ y= -2-t \\ z= 7+5t \end{cases}\; , \; t \in \mathbb{R} .$$