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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 $y = x^{2} + x + 10$ at a point $A (x_{0}, y_{0}),$ where $x_{0} = 1.$ The tangent cuts the Ox axis at a point B. Find the scalar product of the vectors $- - \to O A a n d - - \to A R .$

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Solution

Ordinate of A is given by substituting the value of $x_{0}$ in the curve equation. Thus $y_{0} = 1^{2} + 1 + 10 = 12$ .

The slope of any tangent at A is $m = 2 x_{0} + 1 = 3$ . The equation of tangent at the point A is given by,

$y = 12 + 3 (x - 1) = 3 x + 9$

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, $B = (- 3, 0)$

$\to A B = \to O B - \to O A = - 3 i - i - 12 j = - 4 i - 12 j$

$\to O A . \to A B = (i + 12 j) . (- 4 i - 12 j) = - 4 - 144 = - 148$

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