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Question

Derive the equation of the line in space passing through a point and parallel to a vector both in vector and Cartesian form.

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Solution

A line is determined by a point and a direction. Thus, to find an equation representing a line in three dimensions choose a point P0 on the line and a non-zero vector v parallel to the line.
Since any constant multiple of a vector still points in the same direction, it seems reasonable that a point on the line can be found by starting at the point P0 on the line and following a constant multiple of the vector v (see the figure above).
If r is the position vector of P, then the line must be in the form
r=r0+tv

Let the direction ratios of the line be a,b,c. Consider coordinates of any point A be (x,y,z).
Then Cartesian form of a line is given by
xx0a=yy0b=zz0c

658074_623556_ans_40c37ba686cb427992cb63eb97b68a44.png

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