Question

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and .

Answer 1

The line is passing through the point ( 1,2,3 ).

The given equations of the planes are,

r → ⋅( i ^ − j ^ +2 k ^ )=5(1)

r → ⋅( 3 i ^ + j ^ + k ^ )=6(2)

Let b → be the vector. The equation of line which is parallel to the vector is given by,

b → = b → 1 i ^ + b → 2 j ^ + b → 3 k ^

The position of the vector of point ( 1,2,3 ) is given by,

a → = i ^ +2 j ^ +3 k ^

The formula for the equation of line passing through the point ( x 1 , y 1 , z 1 ) and parallel to vector a 1 i ^ + b 1 j ^ + c 1 k ^ is given by,

a → +λ b →

Substitute the values in the above equation,

r → ⋅( i ^ + j ^ +3 k ^ )+λ( b → 1 i ^ + b → 2 j ^ + b → 3 k ^ )(3)

According to the given condition, line (1) and plane (3) are parallel to each other,

( i ^ − j ^ +2 k ^ )+λ( b → 1 i ^ + b → 2 j ^ + b → 3 k ^ )=0 λ( b 1 − b 2 +2 b 3 )=0 b 1 − b 2 +2 b 3 =0 (4)

Similarly, line (2) and line (3) are perpendicular to each other,

( 3 i ^ + j ^ + k ^ )+λ( b → 1 i ^ + b → 2 j ^ + b → 3 k ^ )=0 λ( 3b 1 + b 2 + b 3 )=0 3b 1 + b 2 + b 3 =0 (5)

From the equation (4) and equation (5),

b 1 ( −1 )( 1 )−1×2 = b 2 ( 2×3 )−1×1 = b 3 1×1−3×−1 b 1 −3 = b 2 5 = b 3 4

The direction of b → are −3,5 and 4.

b → =−3 i ^ +5 j ^ +4 k ^

Substitute the values in equation (3).

r → ⋅( i ^ +2 j ^ −4 k ^ )+λ( −3 i ^ +5 j ^ +4 k ^ )

Thus, the equation of line passing through the point ( −3,5,4 ) is r → ⋅( i ^ +2 j ^ −4 k ^ )+λ( −3 i ^ +5 j ^ +4 k ^ ).