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A line passing through the point A with position vector a=4i+2j+2k is parallel to the vector b=2i+3j+6k . Find the length of the perpendicular drawn on this line from a point P with vector r1=i+2j+3k - CBSE (Commerce) Class 12 - Mathematics

Question

A line passing through the point A with position vector veca=4hati+2hatj+2hatk is parallel to the vector vecb=2hati+3hatj+6hatk . Find the length of the perpendicular drawn on this line from a point P with vector vecr_1=hati+2hatj+3hatk

Solution

Let the equation of the line be vecr=veca+lambda vecb

Here  veca=4hati+2hatj+2hatk and vecb=2hati+3hatj+6hatk

equation of the line =4hati+2hatj+2hatk+lambda (2hati+3hatj+6hatk)

Let L be the foot of the perpendicular and P be the required point from which we have to find the length of the perpendicular

P(vecalpha)=hati+2hatj+3hatk

vec(PL)=position vector of L -position vector of P

=4hati+2hatj+2hatk+lambda (2hati+3hatj+6hatk)-(hati+2hatj+3hatk)

=3hati-hatk+lambda(2hati+3hatj+6hatk).................(i)

Now, vec(PL).vecb=0[Since vec(PL)" is perpendicular to "vecb]

=3hati-hatk+lambda(2hati+3hatj+6hatk).(2hati+3hatj+6hatk)=0

=>(3+2lambda)2+(3lambda)3+(-1+6lambda)6=0

=>6+4lambda+9lambda-6+36lambda=0

=>49lambda=0

therefore lambda=0

vec(PL)=3hati-hatk ["from "(ii)]

|vec(PL)|=|sqrt(3^2+(-1)^2)|

therefore  |vec(PL)|=sqrt(10)

Length of the perpendicular drawn on the line from P=sqrt(10)

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Solution A line passing through the point A with position vector a=4i+2j+2k is parallel to the vector b=2i+3j+6k . Find the length of the perpendicular drawn on this line from a point P with vector r1=i+2j+3k Concept: Vectors Examples and Solutions.
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