Find the equation of the line in vector and in Cartesian form that passes through the point with position vector and is in the direction .

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Solution

It is given that the line passes through the point having position vector ( 2 i ^ − j ^ +4 k ^ ) and is in the direction ( i ^ +2 j ^ − k ^ ).

Consider that the given line passes through the point with position vector m → and is parallel to vector n → .

The position vectors are,

m → =2 i ^ − j ^ +4 k ^ n → = i ^ +2 j ^ − k ^

The equation of the line passing through a point is given as,

r → = m → +λ n → (1)

Substitute the values of m → and n → in equation (1).

r → =( 2 i ^ − j ^ +4 k ^ )+λ( i ^ +2 j ^ − k ^ ) =( λ+2 ) i ^ +( 2λ−1 ) j ^ +( −λ+4 ) k ^ (2)

Consider the equation of line in Cartesian form as,

r → =x i ^ +y j ^ +z k ^ (3)

Compare equation (2) and (3),

x i ^ +y j ^ +z k ^ =( λ+2 ) i ^ +( 2λ−1 ) j ^ +( −λ+4 ) k ^ x=λ+2 y=2λ−1 z=−λ+4

The equation of the given line in Cartesian form by eliminating λ is,

x−2 1 = y+1 2 = z−4 −1

Thus, the required equation of the line in Cartesian form is x−2 1 = y+1 2 = z−4 −1 .

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