The Cartesian equation of a line is . Write its vector form.
The given equation of the line in Cartesian form is,
x − 5 3 = y + 4 7 = z − 6 2 x − 5 3 = y − ( − 4 ) 7 = z − 6 2 (1)
The equation of the line in Cartesian form is represented as,
x − x ′ a = y − y ′ b = z − z ′ c (2)
Compare both the equations (1) and (2).
x ′ = 5 , y ′ = − 4 , z ′ = 6 a = 3 , b = 7 , c = 2
The vector form of the line equation is,
r → = a → + λ b → (3)
Where,
a → = x ′ i ^ + y ′ j ^ + z ′ k ^ , b → = a i ^ + b j ^ + c k ^ .
Substitute the values in a → , b → .
a → = 5 i ^ − 4 j ^ + 6 k ^ , b → = 3 i ^ + 7 j ^ + 2 k ^
Substitute the value of a → and b → in equation (3).
r → = ( 5 i ^ − 4 j ^ + 6 k ^ ) + λ ( 3 i ^ + 7 j ^ + 2 k ^ )
Thus, the equation of the line in vector form is r → = ( 5 i ^ − 4 j ^ + 6 k ^ ) + λ ( 3 i ^ + 7 j ^ + 2 k ^ ) .
The given equation of the line in Cartesian form is,
The equation of the line in Cartesian form is represented as,
Compare both the equations (1) and (2).
The vector form of the line equation is,
Where,
Substitute the values in
Substitute the value of
Thus, the equation of the line in vector form is
