Question

Calculate the scalar product of the following vectors.
A point A $(x_1,y_1)$ with abscissa $x_1=1$ and a point B $(x_2,y_2)$ with ordinate $y_2=1$ are given in the rectangular Cartesian system of coordinates Oxy on the curve $y=x^2-4x+5.$ Find the scalar product of the vectors $\overrightarrow{OA}and\overrightarrow{OB}.$

Answer 1

The ordinate of the point A is found by substituting the value $x_1=1$ in the curve equation. This yields us $y_1=1^2-4\left(1\right)+5=2$

The abscissa of the point B is found by substituting the value $y_2=1$ in the curve equation. This yields us $1=x_2^2-4x_2+5$

$⟹x_2^2-4x_2+4=0$.

solving this quadratic equation gives us, $x_2=2$.

thus $A=(1,2)$ and $B=(2,1)$

$\overrightarrow{OA}.\overrightarrow{OB}=1.2+2.1=4$