Question

Find the vector and Cartesian equation of the planes (a) that passes through the point (1, 0, −2) and the normal to the plane is . (b) that passes through the point (1, 4, 6) and the normal vector to the plane is .

Answer 1

Consider the expression of a vector with coordinates ( a 0 , b 0 , c 0 ) lying in the plane,

r → 0 = a 0 i ^ + b 0 j ^ + c 0 k ^ (1)

The vector equation of the plane when the coordinates ( a 0 , b 0 , c 0 )lie on the plane is,

( r → − r → 0 )⋅ n → =0(2)

Here,

n → is the vector normal to the plane.

The normal vector is expressed as,

n → =A i ^ +B j ^ +C k ^ (3)

The vector equation of the plane when the coordinates ( a 0 , b 0 , c 0 )lie on the plane is,

A( x− a 0 )+B( y− b 0 )+C( z− c 0 )=0(4)

(a)

If the plane passes through the point ( 1,0,−2 ) , then comparing with standard notation,

a 0 =1

b 0 =0

c 0 =−2

Substitute the values in equation (1).

r → 0 = i ^ −2 k ^ (5)

The normal to the vector is given as,

n → = i ^ + j ^ − k ^ (6)

Compare equation (6) and (3).

A=1

B=1

C=−1

Substitute the values in equation (2) to obtain the vector equation.

( r → −( i ^ −2 k ^ ) )⋅( i ^ + j ^ − k ^ )=0

Substitute the values in equation (4) to obtain the vector equation.

( x−1 )+y−( z+2 )=0 x+y−z−1−2=0 x+y−z=3

Hence, the vector equation of the plane is ( r → −( i ^ −2 k ^ ) )⋅( i ^ + j ^ − k ^ )=0and the Cartesian equation of the plane is x+y−z=3.

(b)

If the plane passes through the point ( 1,4,6 ) , then comparing with standard notation

a 0 =1

b 0 =4

c 0 =6

Substitute the values in equation (1).

r → 0 = i ^ +4 j ^ +6 k ^ (5)

The normal to the vector is given as,

n → = i ^ −2 j ^ + k ^ (6)

Compare equation (6) and (3).

A=1

B=−2

C=1

Substitute the values in equation (2) to obtain the vector equation.

( r → −( i ^ +4 j ^ +6 k ^ ) )⋅( i ^ −2 j ^ + k ^ )=0

Substitute the values in equation (4) to obtain the vector equation.

( x−1 )−2( y−4 )+( z−6 )=0 x−2y−z−1+8−6=0 x−2y+z+1=0

Hence, the vector equation of the plane is ( r → −( i ^ +4 j ^ +6 k ^ ) )⋅( i ^ −2 j ^ + k ^ )=0and the Cartesian equation of the plane is x−2y+z+1=0.