Digest for "The Canonical Vector Plane"

For a vector \( \overrightarrow{u}=\begin{bmatrix}a\\ b\end{bmatrix}\) , where \( (a,b)\in(\mathbb{R}_+^*)^2\) are real numbers such as \( a>0\) and \( b>0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}=a\) is the distance between the origin and the projection on the \( x\) axis along the \( y\) axis of the end of the vector \( \overrightarrow{u}\) .

• \( y_{u}=b\) is the distance between the origin and the projection on the \( y\) axis along the \( x\) axis of the end of the vector \( \overrightarrow{u}\) .

For a vector \( \overrightarrow{u}=\begin{bmatrix}a\\ b\end{bmatrix}\) , where \( (a,b)\in\mathbb{R}_-^*\times\mathbb{R}_+^*\) are real numbers such as \( a<0\) and \( b>0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}=a\) is the opposite of the distance between the origin and the projection on the \( x\) axis along the \( y\) axis of the end of the vector \( \overrightarrow{u}\) .

• \( y_{u}=b\) is the distance between the origin and the projection on the \( y\) axis along the \( x\) axis of the end of the vector \( \overrightarrow{u}\) .

For a vector \( \overrightarrow{u}=\begin{bmatrix}a\\ b\end{bmatrix}\) , where \( (a,b)\in(\mathbb{R}_-^*)^2\) are real numbers such as \( a<0\) and \( b<0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}=a\) is the opposite of the distance between the origin and the projection on the \( x\) axis along the \( y\) axis of the end of the vector \( \overrightarrow{u}\) .

• \( y_{u}=b\) is the opposite of the distance between the origin and the projection on the \( y\) axis along the \( x\) axis of the end of the vector \( \overrightarrow{u}\) .

For a vector \( \overrightarrow{u}=\begin{bmatrix}a\\ b\end{bmatrix}\) , where \( (a,b)\in\mathbb{R}_+^*\times\mathbb{R}_-^*\) are real numbers such as \( a>0\) and \( b<0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}=a\) is the distance between the origin and the projection on the \( x\) axis along the \( y\) axis of the end of the vector \( \overrightarrow{u}\) .

• \( y_{u}=b\) is the opposite of the distance between the origin and the projection on the \( y\) axis along the \( x\) axis of the end of the vector \( \overrightarrow{u}\) .

For a vector \( \overrightarrow{u}=\begin{bmatrix}a\\ 0\end{bmatrix}\) , where \( a\in\mathbb{R}_+^*\) is a real number such as \( a>0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}=a\) is the distance between the origin and the end of the vector \( \overrightarrow{u}\) , that is the length of the vector \( \overrightarrow{u}\) .

• \( y_{u}\) is equal to \( 0\) .

For a vector \( \overrightarrow{u}=\begin{bmatrix}0\\ b\end{bmatrix}\) , where \( b\in\mathbb{R}_+^*\) is a real number such as \( b>0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}\) is equal to \( 0\)

• \( y_{u}=b\) is the distance between the origin and the end of the vector \( \overrightarrow{u}\) , that is the length of the vector \( \overrightarrow{u}\) . .

For a vector \( \overrightarrow{u}=\begin{bmatrix}a\\ 0\end{bmatrix}\) , where \( a\in\mathbb{R}_-^*\) is a real number such as \( a<0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}=a\) is the opposite of the distance between the origin and the end of the vector \( \overrightarrow{u}\) , that is the opposite of the length of the vector \( \overrightarrow{u}\) .

• \( y_{u}\) is equal to \( 0\) .

For a vector \( \overrightarrow{u}=\begin{bmatrix}0\\ b\end{bmatrix}\) , where \( b\in\mathbb{R}_-^*\) is a real number such as \( b<0\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}\) is equal to \( 0\) .

• \( y_{u}=b\) is the opposite of the distance between the origin and the end of the vector \( \overrightarrow{u}\) , that is the opposite of the length of the vector \( \overrightarrow{u}\) .

For the null vector \( \overrightarrow{u}=\overrightarrow{0}=\begin{bmatrix}0\\ 0\end{bmatrix}\) , its abscissa \( x_{u}\) and its ordinate \( y_{u}\) are the following:

• \( x_{u}\) is equal to \( 0\) .

• \( y_{u}\) is equal to \( 0\) as well.

Assume \( (x,y,z,t)\in\mathbb{R}^4\) are real numbers, and consider the two column vectors with 2 elements, \( \overrightarrow{u}=\begin{bmatrix}x\\ y\end{bmatrix}\) and \( \overrightarrow{v}=\begin{bmatrix}z\\ t\end{bmatrix}\) .

Then we define the sum and the difference of \( \overrightarrow{u}\) and \( \overrightarrow{v}\) the following way:

• \( \overrightarrow{u}+\overrightarrow{v}=\begin{bmatrix}x+z\\ y+t\end{bmatrix}\) is the sum element by element of \( \overrightarrow{u}\) and \( \overrightarrow{v}\) ,

• \( \overrightarrow{u}-\overrightarrow{v}=\begin{bmatrix}x-z\\ y-t\end{bmatrix}\) is the difference element by element of \( \overrightarrow{u}\) and \( \overrightarrow{v}\)

Assume \( (x_1,y_1,x_2,y_2,x_3,y_3)\in\mathbb{R}^6\) are real numbers, and consider the three column vectors with 2 elements, \( \overrightarrow{u}_1=\begin{bmatrix}x_1\\ y_1\end{bmatrix}\) , \( \overrightarrow{u}_2=\begin{bmatrix}x_2\\ y_2\end{bmatrix}\) and \( \overrightarrow{u}_3=\begin{bmatrix}x_3\\ y_3\end{bmatrix}\) .

Then the following assertions hold:

• \( \overrightarrow{u}_1+\overrightarrow{u}_2=\overrightarrow{u}_2+\overrightarrow{u}_1\) : the addition of vectors is commutative.

• \( (\overrightarrow{u}_1+\overrightarrow{u}_2)+\overrightarrow{u}_3=\overrightarrow{u}_1+(\overrightarrow{u}_2+\overrightarrow{u}_3)\) : the addition of vectors is associative.

• \( \overrightarrow{u_{1}}+\overrightarrow{0}=\overrightarrow{0}+\overrightarrow{u_{1}} =\overrightarrow{u_{1}}\) : the null vector \( \overrightarrow{0}=\begin{bmatrix}0\\ 0\end{bmatrix}\) is neutral for the addition of vectors.

• The opposite \( -\overrightarrow{u_{1}}=\begin{bmatrix}-x_{1}\\ {-y_{1}}\end{bmatrix}\) of \( \overrightarrow{u_{1}}\) is its reciprocal for the addition of vectors: \( \overrightarrow{u_{1}}+(-\overrightarrow{u_{1}}) =(-\overrightarrow{u_{1}})+\overrightarrow{u_{1}}=\overrightarrow{0}\)

Assume \( (x,y)\in\mathbb{R}^2\) are real numbers, and consider the column vector \( \overrightarrow{u}=\begin{bmatrix}x\\ y\end{bmatrix}\) and its opposite \( -\overrightarrow{u}=\begin{bmatrix}-x\\ {-y}\end{bmatrix}\) .

Then the following assertions hold:

• \( -\overrightarrow{u}\) is aligned with \( \overrightarrow{u}\) ,

• \( -\overrightarrow{u}\) is in the direction opposite to the direction of \( \overrightarrow{u}\) ,

• and \( \left\|-\overrightarrow{u}\right\|=\left\|\overrightarrow{u}\right\|\) .

Assume \( (\overrightarrow{u},\overrightarrow{v})\in\mathbb{P}^2\) are column vectors with two real elements.

Then the following assertion holds:

Assume \( (\overrightarrow{u},\overrightarrow{v})\in\mathbb{P}^2\) are column vectors with two real elements.

Then the following assertions hold:

• \( (\overrightarrow{u}+\overrightarrow{v})-\overrightarrow{v}=\overrightarrow{u}\)

• \( (\overrightarrow{u}-\overrightarrow{v})+\overrightarrow{v}=\overrightarrow{u}\)

Assume \( (\overrightarrow{u},\overrightarrow{v},\overrightarrow{w})\in\mathbb{P}^3\) are column vectors with two real elements.

Then the following equivalences hold:

1. \( \overrightarrow{w}=\overrightarrow{u}+\overrightarrow{v} \Leftrightarrow \overrightarrow{u}=\overrightarrow{w}-\overrightarrow{v}\)

2. \( \overrightarrow{w}=\overrightarrow{u}-\overrightarrow{v} \Leftrightarrow \overrightarrow{u}=\overrightarrow{w}+\overrightarrow{v}\)

3. \( \overrightarrow{w}=\overrightarrow{v}-\overrightarrow{u} \Leftrightarrow \overrightarrow{u}=\overrightarrow{v}-\overrightarrow{w}\)

Assume \( (x,y)\in\mathbb{R}^2\) are real numbers, and consider the column vector \( \overrightarrow{u}=\begin{bmatrix}x\\ y\end{bmatrix}\) with 2 elements.

Assume \( \lambda\in\mathbb{R}\) is a real number.

Then we multiply and divide the vector \( \overrightarrow{u}\) by the scalar \( \lambda\) the following way:

• \( \lambda\overrightarrow{u}=\begin{bmatrix}\lambda x\\ \lambda y\end{bmatrix}\) is the product element by element of \( \overrightarrow{u}\) by \( \lambda\) ,

• and, provided \( \lambda\neq 0\) , \( \frac{\overrightarrow{u}}{\lambda}=\begin{bmatrix}\frac{x}{\lambda}\\ \frac{y}{\lambda}\end{bmatrix}\) is the quotient element by element of \( \overrightarrow{u}\) by \( \lambda\) .

Assume \( \overrightarrow{u}\in\mathbb{P}^*\) is a column vector with two real elements such as \( \overrightarrow{u}\neq\overrightarrow{0}\) .

Then the following assertions hold:

• If we multiply the vector \( \overrightarrow{u}\) by the scalar \( 1\) , we obtain the vector \( \overrightarrow{u}\) : \( 1.\overrightarrow{u}=\overrightarrow{u}\) .

• If we divide the vector \( \overrightarrow{u}\) by the scalar \( 1\) , we obtain the vector \( \overrightarrow{u}\) : \( \frac{\overrightarrow{u}}{1}=\overrightarrow{u}\) .

Assume \( \overrightarrow{u}\in\mathbb{P}^*\) is a column vector with two real elements such as \( \overrightarrow{u}\neq\overrightarrow{0}\) .

Then the following assertions hold:

• If we multiply the vector \( \overrightarrow{u}\) by the scalar \( -1\) , we obtain the opposite \( -\overrightarrow{u}\) of the vector \( \overrightarrow{u}\) : \( (-1).\overrightarrow{u}=-\overrightarrow{u}\) .

• If we divide the vector \( \overrightarrow{u}\) by the scalar \( -1\) , we obtain the opposite \( -\overrightarrow{u}\) of the vector \( \overrightarrow{u}\) : \( \frac{\overrightarrow{u}}{-1}=-\overrightarrow{u}\) .

Assume \( \overrightarrow{u}\in\mathbb{P}^*\) is a column vector with two real elements such as \( \overrightarrow{u}\neq\overrightarrow{0}\) , and \( \lambda\in\mathbb{R}_+^*\) is a real number such as \( \lambda>0\) .

Then, if we multiply the vector \( \overrightarrow{u}\) by the scalar \( \lambda\) , we obtain a vector
\( \overrightarrow{v}=\lambda\overrightarrow{u}\) such as:

• \( \overrightarrow{v}\) is aligned with \( \overrightarrow{u}\) ,

• and \( \overrightarrow{v}\) is in the same direction as \( \overrightarrow{u}\) .

Assume \( \overrightarrow{u}\in\mathbb{P}^*\) is a column vector with two real elements such as \( \overrightarrow{u}\neq\overrightarrow{0}\) , and \( \lambda\in\mathbb{R}_+^*\) is a real number such as \( \lambda>0\) .

Then, if we divide the vector \( \overrightarrow{u}\) by the scalar \( \lambda\) , we obtain a vector \( \overrightarrow{v}=\frac{\overrightarrow{u}}{\lambda}\) such as:

• \( \overrightarrow{v}\) is aligned with \( \overrightarrow{u}\) ,

• \( \overrightarrow{v}\) is in the same direction as \( \overrightarrow{u}\) ,

• and \( \overrightarrow{v}=\frac{1}{\lambda}\overrightarrow{u}\) .

Assume \( \overrightarrow{u}\in\mathbb{P}^*\) is a column vector with two real elements such as \( \overrightarrow{u}\neq\overrightarrow{0}\) , and \( \lambda\in\mathbb{R}_-^*\) is a real number such as \( \lambda<0\) .

Then, if we multiply the vector \( \overrightarrow{u}\) by the scalar \( \lambda\) , we obtain a vector
\( \overrightarrow{v}=\lambda\overrightarrow{u}\) such as:

• \( \overrightarrow{v}\) is aligned with \( \overrightarrow{u}\) ,

• \( \overrightarrow{v}\) is in the direction opposite to the direction of \( \overrightarrow{u}\) ,

• and \( \overrightarrow{v}\) is the opposite \( -\left|\lambda\right|\overrightarrow{u}\) of the product of \( \overrightarrow{u}\) by the absolute value of \( \lambda\) .

Assume \( \overrightarrow{u}\in\mathbb{P}^*\) is a column vector with two real elements such as \( \overrightarrow{u}\neq\overrightarrow{0}\) , and \( \lambda\in\mathbb{R}_-^*\) is a real number such as \( \lambda<0\) .

Then, if we divide the vector \( \overrightarrow{u}\) by the scalar \( \lambda\) , we obtain a vector \( \overrightarrow{v}=\frac{\overrightarrow{u}}{\lambda}\) such as:

• \( \overrightarrow{v}\) is aligned with \( \overrightarrow{u}\) ,

• \( \overrightarrow{v}\) is in the opposite direction of the direction of \( \overrightarrow{u}\) ,

• and \( \overrightarrow{v}\) is the opposite \( -\frac{1}{\left|\lambda\right|}\overrightarrow{u}\) of the product of \( \overrightarrow{u}\) by the inverse of the absolute value of \( \lambda\) .

Assume \( \overrightarrow{u}\in\mathbb{P}\) is a column vector with two real elements.

Then the following assertions hold:

• If we multiply the vector \( \overrightarrow{u}\) by the scalar \( 0\) , we obtain the null vector \( \overrightarrow{0}\) : \( (0).\overrightarrow{u}=\overrightarrow{0}\) .

• The vector \( \overrightarrow{u}\) can not be divided by the scalar \( 0\) .

Assume \( \lambda\in\mathbb{R}\) is a real number.

Then the following assertions hold:

• If we multiply the null vector \( \overrightarrow{0}\) by the scalar \( \lambda\) , we obtain the null vector \( \overrightarrow{0}\) : \( (0).\overrightarrow{u}=\overrightarrow{0}\) .

• If \( \lambda\ne 0\), and if we divide the null vector \( \overrightarrow{0}\) by the non zero scalar \( \lambda\) , we obtain the null vector \( \overrightarrow{0}\) : \( \frac{\overrightarrow{0}}{\lambda}=\overrightarrow{0}\) .

Assume \( (x,y)\in\mathbb{R}^2\) are real numbers, and consider the column vector \( \overrightarrow{u}=\begin{bmatrix}x\\ y\end{bmatrix}\) with 2 elements.

Consider the canonical base of the vector plane \( \mathbb{P}\) : \( \overrightarrow{i}=\begin{bmatrix}1\\ 0\end{bmatrix}\) and \( \overrightarrow{j}=\begin{bmatrix}0\\ 1\end{bmatrix}\) .

Then the coordinates of \( \overrightarrow{u}\) in the canonical base are:

• its abscissa \( x\) ,

• and its ordinate \( y\) .

Moreover, the following identity holds:

• \( \overrightarrow{u}=x\overrightarrow{i}+y\overrightarrow{j}\) .

Assume \( \overrightarrow{u}\in\mathbb{P}\) is a column vector with two real elements, and \( \lambda\in\mathbb{R}^*\) is a real number such as \( \lambda\neq 0\) .

Then the following assertions hold:

• \( \lambda\frac{\overrightarrow{u}}{\lambda}=\overrightarrow{u}\) ,

• and \( \frac{\lambda\overrightarrow{u}}{\lambda}=\overrightarrow{u}\) .

Assume \( \overrightarrow{u}\in\mathbb{P}\) is a column vector with two real elements, and \( \lambda\in\mathbb{R}^*\) is a real number such as \( \lambda\neq 0\) .

Then the following assertion holds:

• \( \frac{\overrightarrow{u}}{\lambda}=\frac{1}{\lambda}\overrightarrow{u}\) .

Assume \( \overrightarrow{u}\in\mathbb{P}\) is a column vector with two real elements, and \( (\alpha,\beta)\in\mathbb{R}^{2}\) is are real numbers.

Then the following associativity property holds:

• \( \alpha(\beta\overrightarrow{u})=(\alpha\beta)\overrightarrow{u}\)

Assume \( (\overrightarrow{u},\overrightarrow{v})\in\mathbb{P}^2\) are column vectors with two real elements, and \( \lambda\in\mathbb{R}^*\) is a real number such as \( \lambda\neq 0\) .

Then the following equivalences hold:

• \( \overrightarrow{v}=\lambda\overrightarrow{u} \Leftrightarrow \overrightarrow{u}=\frac{\overrightarrow{v}}{\lambda}\) ,

• and \( \overrightarrow{v}=\frac{\overrightarrow{u}}{\lambda} \Leftrightarrow \overrightarrow{u}=\lambda\overrightarrow{v}\)

Assume \( (\overrightarrow{u},\overrightarrow{v})\in\mathbb{P}^2\) are column vectors with two real elements, and \( (\alpha,\beta)\in\mathbb{R}^2\) are real numbers.

Then the following assertions hold:

• First distributivity law: \( \alpha(\overrightarrow{u}+\overrightarrow{v})=\alpha\overrightarrow{u}+\alpha\overrightarrow{v}\)

• Second distributivity law: \( (\alpha+\beta)\overrightarrow{u}=\alpha\overrightarrow{u}+\beta\overrightarrow{u}\)

• Associativity law: \( \alpha(\beta\overrightarrow{u})=(\alpha\beta)\overrightarrow{u}\)

Assume \( (\overrightarrow{u},\overrightarrow{v})\in\mathbb{P}^2\) are column vectors with two real elements, and \( (\alpha,\beta)\in\mathbb{R}^2\) are real numbers.

Then the following assertions hold:

• Signs law: \( (-\alpha)\overrightarrow{u}=\alpha(-\overrightarrow{u})=-\alpha\overrightarrow{u}\)

• First distributivity law: \( \alpha(\overrightarrow{u}-\overrightarrow{v})=\alpha\overrightarrow{u}-\alpha\overrightarrow{v}\)

• Second distributivity law: \( (\alpha-\beta)\overrightarrow{u}=\alpha\overrightarrow{u}-\beta\overrightarrow{u}\)

Assume \( \lambda\in\mathbb{R}\) is a real number, and consider the homothety of factor \( \lambda\) in \( \mathbb{P}\) :

Assume \( (\overrightarrow{u},\overrightarrow{v})\in\mathbb{P}^2\) are column vectors with two real elements, and \( (\overrightarrow{u},\overrightarrow{v})\in\mathbb{P}^2\) is a real number.

Then the following assertions hold:

• \( h_{\lambda}(\overrightarrow{u}+\overrightarrow{v}) =h_{\lambda}(\overrightarrow{u})+h_{\lambda}(\overrightarrow{v})\) ,

• and \( h_{\lambda}(\alpha \overrightarrow{u})=\alpha h_{\lambda}(\overrightarrow{u})\) .

Assume \( (\lambda,\mu)\in\mathbb{R}^2\) are real numbers, and consider the
homotheties of factors \( \lambda\) and \( \mu\) in \( \mathbb{P}\) :

and

Assume \( \overrightarrow{u}\in\mathbb{P}\) is a column vector with two real elements.

Then the following assertions hold:

• \( h_{\lambda+\mu}(\overrightarrow{u})=h_{\lambda} (\overrightarrow{u})+h_{\mu}(\overrightarrow{u})\) ,

• and \( h_{\lambda\mu}(\overrightarrow{u})=h_{\lambda}(h_{\mu}(\overrightarrow{u}))\) .