Question

Calculate the scalar product of the following vectors.
Two points A and B are given in the rectangular Cartesian system of coordinates Oxy on the curve $y=2^{x+2},$ the points being such that $\overrightarrow{OA}\cdot i=-1and\overrightarrow{OB}\cdot i=2,$ where 1 is a unit vector of the Ox axis. Find the length of the vector $\overrightarrow{-4OA}+\overrightarrow{OB}.$

Answer 1

let $A=(x_1,y_1)$ and $B=(x_2,y_2)$

given $\overrightarrow{OA}.i=-1⟹x_1=-1$ and $\overrightarrow{OB}.i=2⟹x_2=2$

$A,B$ are points on the curve $y=2^{x+2}$. Using this to find the ordinates of A and B.

$y_1=2^{x_1+2}=2$ and $y_2=2^{x_2+2}=16$

$-4\overrightarrow{OA}+\overrightarrow{OB}=-4(-i+2j)+(2i+16j)=6i+8j$

The length of this vector is given by $\sqrt{6^2+8^2}=\sqrt{100}=10$