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Question

Find the equation of line in vector and in cartesian form that passes through the point with position vector 2i^-j^+4k^ and is in the direction i^+2j^-k^


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Solution

Step 1: Solve for the equation of the line in vector form

It is given that the line passes through the point with position vector a=2i^-j^+4k^ and in the direction b=i^+2j^-k^

Let r is any line vector
Equation of a line through a point with position vector aand parallel to bis

r=a+λb where λ is an arbitrary constant

So the required equation of the line in vector form is

r=2i^-j^+4k^+λi^+2j^-k^

(λ+2)i^+(2λ-1)j^+(-λ+4)k^

Equation of line in vector form is 2+λi^+2λ-1j^+4-λk^

Step 2: Solve for the equation of the line in cartesian form

General equation of line in cartesian form passing through point x1,y1,z1 and having direction ratio a,b,c is

x-x1a=y-y1b=z-z1c

Since the line passes through a point with position vector 2i^-j^+4k^

x1=2,y1=-1,z1=4

Also line is in the direction of i^+2j^-k^

Direction ratios are a=1,b=2,c=-1

x-21=y-(-1)2=z-4-1x-21=y+12=z-4-1

Hence the equations of line in vector and cartesian form are (2+λ)i^+(2λ-1)j^+(4-λ)k^ and x-21=y+12=z-4-1 respectively.


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