Question

Find the vector and the Cartesian equations of the line that passes through the points (3, −2, −5), (3, −2, 6).

Answer 1

It is given that the line passes through the points ( 3,−2,−5 ) and ( 3,−2,6 ).

The vector equation of the line passing through two given points with position vectors m →  and  n → is represented by,

r → = m → +λ( n → − m → ) (1)

Given two position vectors are,

m → =3 i ^ −2 j ^ −5 k ^ n → =3 i ^ −2 j ^ +6 k ^

Now, substitute the values of m →  and  n → in equation (1).

r → = m → +λ( n → − m → ) r → =( 3 i ^ −2 j ^ −5 k ^ )+λ( 3 i ^ −2 j ^ +6 k ^ −( 3 i ^ −2 j ^ −5 k ^ ) ) r → =( 3 i ^ −2 j ^ −5 k ^ )+λ( 0 i ^ −0 j ^ +11 k ^ ) r → =3 i ^ −2 j ^ −5 k ^ +λ( 11 k ^ )

Thus, the vector form of the given line equation is r → =3 i ^ −2 j ^ −5 k ^ +λ( 11 k ^ ).

The Cartesian equation of the line passing through two points is given by,

x− x 1 x 2 − x 1 = y− y 1 y 2 − y 1 = z− z 1 z 2 − z 1 (2)

As the line passes through two given points then,

m → =( x 1 , y 1 , z 1 ) x 1 =3, y 1 =−2, z 1 =−5 n → =( x 2 , y 2 , z 2 ) x 2 =3, y 2 =−2, z 2 =6

Substitute these values in equation (2).

x−3 3−3 = y−( −2 ) −2−( −2 ) = z−( −5 ) 6−( −5 ) x−3 0 = y+2 0 = z+5 11

Thus, the Cartesian equation of the line is x−3 0 = y+2 0 = z+5 11 .