Question

Find the vector and the Cartesian equations of the lines that pass through the origin and (5, −2, 3).

Answer 1

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

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

r → = m → +λ( n → − m → ) (1)

Given two position vectors are,

m → =0 i ^ +0 j ^ +0 k ^ n → =5 i ^ −2 j ^ +3 k ^

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

r → = m → +λ( n → − m → ) r → =( 0 i ^ +0 j ^ +0 k ^ )+λ( 5 i ^ −2 j ^ +3 k ^ −( 0 i ^ +0 j ^ +0 k ^ ) ) r → =λ( 5 i ^ −2 j ^ +3 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 =0, y 1 =0, z 1 =0 n → =( x 2 , y 2 , z 2 ) x 2 =5, y 2 =−2, z 2 =3

Substitute these values in equation (2).

x−0 5−0 = y−0 −2−0 = z−0 3−0 x 5 = y −2 = z 3

Therefore, the Cartesian equation is x 5 = y −2 = z 3 .