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, which is the point of intersection of the curves $y=2x^2-3x+5andy=2x^2-2x+3,$ are given in the rectangular Cartesian system of coordinates Oxy on the curve $y=2x^2-3x+5.$ 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 $y_1=2\left(1^2\right)-3\left(1\right)+5=4$

Let the point B be $(x_2,y_2)$. It is given to be the intersection of two curves. Thus it must satisfy both the curve equations. Substituting the point in the curve equations yield us $y_2=2x_2^2-3x_2+5$ and $y_2=2x_2^2-2x_2+3$. Eliminating $y_2$ from the two equations gives us

$2x_2^2-3x_2+5=2x_2^2-2x_2+3$

$⟹x_2=2$

Substituting this in the above equation to find $y_2$, we get, $y_2=2(2{)}^2-3(2)+5=7$

thus $A=(1,4)$ and $B=(2,7)$

$\overrightarrow{OA}.\overrightarrow{OB}=1.2+4.7=30$