The dot product is a mathematical operation that measures the angle between two vectors in a vector space. It is defined as the product of the length of the vectors by the cosine of the angle between them. It can be calculated by multiplying the length of one vector by the projection of the other vector onto it. The dot product is a real number that can be positive, negative, or zero.
The vector product is another mathematical operation which allows to create a new vector from two given vectors. It is defined as the product of the length of the vectors by the sine of the angle between them and is oriented in the direction orthogonal to the plane formed by the two vectors. It can be calculated using the right hand rule. The vector product is a vector that can have a length, a direction and a direction.
Dot product and vector product are two different operations that have applications in many mathematical fields such as physics, geometry, mechanics, fluid mechanics, geodesy, etc.
Scalar product
Let be a vector: \( \overrightarrow{\rm u} \left( \begin{array}{cc} x \\ y \end{array} \right) ,\| \overrightarrow{\rm u} \| = \sqrt{x^2+y^2}\)
If a vector is equal to 0 then: \( \overrightarrow{\rm u}.\overrightarrow{\rm v}=0 \)
2 nonzero vectors: \( \overrightarrow{\rm u}.\overrightarrow{\rm v} = \| \overrightarrow{\rm u} \|.\|\overrightarrow{\rm v} \|. cos( \overrightarrow{\rm u},\overrightarrow{\rm v}) \)
If acute angle \( \rightarrow 0 \leq cos(\overrightarrow{\rm u},\overrightarrow{\rm v}) \leq 1 \)
If obtuse angle \( \rightarrow -1 \leq cos(\overrightarrow{\rm u},\overrightarrow{\rm v}) \leq 0 \)
If \( \overrightarrow{\rm u} \) is perpendicular to \( \overrightarrow{\rm v} \) \( \rightarrow cos(\overrightarrow{\rm u},\overrightarrow{\rm v}) = 0 \)
If \( \overrightarrow{\rm u} \) and \( \overrightarrow{\rm v} \) are collinear \( \rightarrow \) in the same direction \( \rightarrow cos(\overrightarrow{\rm u},\overrightarrow{\rm v}) = 1 \) so \( \overrightarrow{\rm u}.\overrightarrow{\rm v} = \|\overrightarrow{\rm u} \|.\|\overrightarrow{\rm v } \| \)
\( \rightarrow \) of opposite direction \( \rightarrow cos(\overrightarrow{\rm u},\overrightarrow{\rm v}) = -1 \) so \( \overrightarrow{\rm u}.\overrightarrow {\rm v} = -\|\overrightarrow{\rm u} \|.\|\overrightarrow{\rm v} \| \)
Standards: \( \overrightarrow{\rm u}.\overrightarrow{\rm v}= \frac{1}{2} *(\|\overrightarrow{\rm u}+\overrightarrow {\rm v} \|^2-\|\overrightarrow{\rm u} \|^2-\|\overrightarrow{\rm v} \|^2 ) \)
Analytically : \( \overrightarrow{\rm u} \left( \begin{array}{cc} x \\ y \end{array} \right) \) and \( \overrightarrow{\ rm v} \left( \begin{array}{cc} x’ \\ y’ \end{array} \right) \rightarrow \overrightarrow{\rm u}.\overrightarrow{\rm v}=x.x’ +y.y’\)
\( \overrightarrow{\rm u} \) and \( \overrightarrow{\rm v} \) orthogonal if their scalar product \( \overrightarrow{\rm u}.\overrightarrow{\rm v} = 0 \rightarrow cos (\overrightarrow{\rm u},\overrightarrow{\rm v}) = 0 \), i.e. \( (\overrightarrow{\rm u},\overrightarrow{\rm v} )= \frac{\pi}{2}\text{ or }\frac{-\pi}{2} \)
\(\overrightarrow{\rm u}.\overrightarrow{\rm v} = 0 \), either \( \overrightarrow{\rm u} = 0 \) or \( \overrightarrow{\rm v} = 0 \) let \( cos(\overrightarrow{\rm u},\overrightarrow{\rm v}) = 0 \)
The null vector \( \overrightarrow{\rm 0} \) orthogonal to any vector
\( \|\overrightarrow{\rm u} \|^2= \overrightarrow{\rm u}.\overrightarrow{\rm u}=\overrightarrow{\rm u}^2 \) (Scalar square of \( \overrightarrow{\rm u}) \)
Properties
Symmetry : \( \overrightarrow{\rm u}.\overrightarrow{\rm v} = \overrightarrow{\rm v}.\overrightarrow{\rm u} \)
Distributivity : \( (\overrightarrow{\rm u}+\overrightarrow{\rm v} ).\overrightarrow{\rm w} = \overrightarrow{\rm u}.\overrightarrow{\rm w}+\overrightarrow{\rm v}.\overrightarrow{\rm w}\)
Multiplication by a real :\( \lambda (\overrightarrow{\rm u}.\overrightarrow{\rm v} ) = ( \lambda . \overrightarrow{\rm u} ).\overrightarrow{ \rm v} \)
\( (a. \overrightarrow{\rm u} ).(b. \overrightarrow{\rm v} ) = a.b.(\overrightarrow{\rm u}.\overrightarrow{\rm v} ) \)
Remarkable identities
$$ (\overrightarrow{\rm u}+\overrightarrow{\rm v} )^2= \overrightarrow{\rm u}^2+\overrightarrow{\rm v}^2 + 2.\overrightarrow{\rm u }.\overrightarrow{\rm v} = \|\overrightarrow{\rm u} \|^2. \| \overrightarrow{\rm v} \|^2 + 2.\overrightarrow{\rm u}.\overrightarrow{\rm v} $$ $$(\overrightarrow{\rm u} – \overrightarrow{\rm v})^2 = \overrightarrow{\rm u}^2 + \overrightarrow{\rm v}^2 – 2.\overrightarrow{\rm u }.\overrightarrow{\rm v} = \|\overrightarrow{\rm u} \|^2. \| \overrightarrow{\rm v} \|^2 – 2.\overrightarrow{\rm u}.\overrightarrow{\rm v}$$ $$(\overrightarrow{\rm u} + \overrightarrow{\rm v} )(\overrightarrow{\rm u} – \overrightarrow{\rm v}) = \overrightarrow{\rm u}^2 – \overrightarrow{ \rm v}^2 = \| \overrightarrow{\rm u} \|^2 – \| \overrightarrow{\rm v} \|^2 $$
Vector product
$$ \| \overrightarrow{\rm u} \wedge \overrightarrow{\rm v} \| = \| \overrightarrow{\rm u} \| . \| \overrightarrow{\rm v} \| .sin(\overrightarrow{\rm u} \wedge \overrightarrow{\rm v})$$
$$ \overrightarrow{\rm u} \left( \begin{array}{cc} x \\ y \\ z \end{array} \right) \text{ et } \overrightarrow{\rm v} \left( \begin{array}{cc} x’ \\ y’ \\ z’ \end{array} \right), \overrightarrow{\rm u} \wedge \overrightarrow{\rm v} \left( \begin{array}{cc} y.z’ – z.y’ \\ z.x’ – x.z’ \\ x.y’ – y.x’ \end{array} \right)$$
3 vectors: \( \overrightarrow{\rm u},\overrightarrow{\rm v},\overrightarrow{\rm w} \)
$$\overrightarrow{\rm D}=\overrightarrow{\rm u} \wedge (\overrightarrow{\rm v} \wedge \overrightarrow{\rm w} ) = (\overrightarrow{\rm u}.\overrightarrow{ \rm w}).\overrightarrow{\rm v} – (\overrightarrow{\rm u}.\overrightarrow{\rm v}).\overrightarrow{\rm w}$$
Distributivity:
$$\overrightarrow{\rm u} \wedge (\overrightarrow{\rm v} + \overrightarrow{\rm w} ) = \overrightarrow{\rm u} \wedge \overrightarrow{\rm v} + \overrightarrow{\ rm u} \wedge \overrightarrow{\rm w}$$
Multiplication by 1 real :
$$\lambda .(\overrightarrow{\rm u} \wedge \overrightarrow{\rm v} )=\lambda .\overrightarrow{\rm u} \wedge \overrightarrow{\rm v}=\overrightarrow{\rm u } \wedge \lambda .\overrightarrow{\rm v}$$
Non-commutativity:
$$\overrightarrow{\rm u} \wedge \overrightarrow{\rm v}=-\overrightarrow{\rm v} \wedge \overrightarrow{\rm u}$$
Determinant:
$$det(\overrightarrow{\rm u},\overrightarrow{\rm v},\overrightarrow{\rm w} ) = det \left( \begin{array}{cc} u_1 & v_1 & w_1 \\ u_2 & v_2 & w_2 \\ u_3 & v_3 & w_3 \end{array} \right) = u_1 .(v_2.w_3 – v_3.w_2 ) – v_1 .(u_2.w_3-u_3.w_2 ) + w_1 .(u_2.v_3 – u_3.v_2)$$