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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=2√x at a point A (x0,y0), where x0=1. The tangent cuts the Ox axis at a point B. Find the scalar product of the vectors −−→ABand−−→OB.
In the rectangular Cartesian system of coordinates Oxy a tangent is drawn to the curve y=2√x at a point A (x0,y0), where x0=1. The tangent cuts the Ox axis at a point B. Find the scalar product of the vectors −−→ABand−−→OB.
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Solution
Ordinate of A is given by substituting the value of x0 in the curve equation. Thus $ y_0=2\sqrt{1}+=2$.
The slope of any tangent at A is m=1√x0=1. The equation of tangent at the point A is given by,
y=2+1(x−1)=x+1
The x−intercept of the tangent is given by substituting y=0 in the tangent line which yields us x=−1 as the x−intercept. So, B=(−1,0)
→AB=→OB−→OA=−i−i−2j=−2i−2j
→OA.→AB=(i+2j).(−2i−2j)=−2−4=−6
Ordinate of A is given by substituting the value of x0 in the curve equation. Thus $ y_0=2\sqrt{1}+=2$.
The slope of any tangent at A is m=1√x0=1. The equation of tangent at the point A is given by,
y=2+1(x−1)=x+1
The x−intercept of the tangent is given by substituting y=0 in the tangent line which yields us x=−1 as the x−intercept. So, B=(−1,0)
→AB=→OB−→OA=−i−i−2j=−2i−2j
→OA.→AB=(i+2j).(−2i−2j)=−2−4=−6
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