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 $y_2=11$ are given in the rectangular Cartesian system of coordinates Oxy on the part of the curve $y=x^2-2x+3$ which lies in the first quadrant. Find the scalar product of $\overrightarrow{OA}and\overrightarrow{OB}.$

Answer 1

The ordinate of the point A is found by substituting the value $x_1=1$ in the curve equation. This yields us $y_1=1^2-2\left(1\right)+3=2$

The abscissa of the point B is found by substituting the value $y_2=11$ in the curve equation. This yields us $11=x_2^2-2x_2+3$

$⟹x_2^2-2x_2-8=0$.

solving this quadratic equation gives us, $x_2=-2,4$.

Since the point is said to be in the first quadrant, we only consider the positive value.

thus $A=(1,2)$ and $B=(4,11)$

$\overrightarrow{OA}.\overrightarrow{OB}=1.4+2.11=26$