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Applications of Dot Product

Calculate the...

Question

Calculate the scalar product of the following vectors.
In the rectangular Cartesian system of coordinates Oxy a tangent is drawn to the curve $y = 8 / x^{2}$ at a point A $(x_{0}, y_{0}),$ where $x_{0} = 2.$ The tangent cuts the Ox axis at a point B. Find the scalar product of the vectors $- - \to A B$ , and $- - \to O B$

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Solution

Ordinate of A is given by substituting the value of $x_{0}$ in the curve equation. Thus $y_{0} = 8 / 2^{2} = 2$ .

The slope of any tangent at A is $m = - 16 / x_{0}^{3} = - 2$ . The equation of tangent at the point A is given by,

$y = 2 - 2 (x - 2) = - 2 x + 6$

The $x - i n t e r c e p t$ of the tangent is given by substituting $y = 0$ in the tangent line which yields us $x = 3$ as the $x - i n t e r c e p t$ . So, $B = (3, 0)$

$\to A B = \to O B - \to O A = 3 i - 2 i - 2 j = - i - 2 j$

$\to O A . \to A B = (2 i + 2 j) . (- i - 2 j) = - 2 - 4 = - 6$

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