Question

13. In the following cases, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them.(a) 7x + 5y + 6z +30 0 and 3x - y -10z 4-0(b) 2r + y 3z-2-0 and x - 2y +5 0(c) 2r- 2y +4z +5-0 and 3x - 3y 6z-1 0(d) 2r - y 3z-10 and 2x -y +3z+3-0(e) 4x + 8y +z- 8-0 and y +z- 4 0

Answer 1

(a)

The equations of the two given planes are,

7x+5y+6z+30=0

And,

3x−y−10z+4=0

Let a 1 , b 1 and c 1 be the direction ratios of the first plane. So,

a 1 =7 b 1 =5 c 1 =6

Let a 2 , b 2 and c 2 be the direction ratios of the second plane. So,

a 2 =3 b 2 =−1 c 2 =−10

If the planes are perpendicular, then,

a 1 a 2 + b 1 b 2 + c 1 c 2 =0 7( 3 )+5( −1 )+6( −10 )=0 21−5−60=0 −44≠0

This shows that the planes are not perpendicular.

If the planes are parallel, then,

a 1 a 2 = b 1 b 2 = c 1 c 2 7 3 = 5 −1 = 6 −10 7 3 ≠−5≠ 3 −5

This shows that the planes are not parallel.

The angle between the planes is,

θ= cos −1 | a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 ⋅ a 2 2 + b 2 2 + c 2 2 | = cos −1 | 7( 3 )+5( −1 )+6( −10 ) 7 2 + 5 2 + 6 2 × 3 2 + ( −1 ) 2 + ( −10 ) 2 | = cos −1 | 21−5−60 49+25+36 × 9+1+100 | = cos −1 | −44 110 × 110 |

Solve further,

θ= cos −1 ( 44 110 ) = cos −1 ( 2 5 )

Therefore, the angle between the planes is cos −1 ( 2 5 ).

(b)

The equations of the two given planes are,

2x+y+3z−2=0

And

x−2y+5=0

Let a 1 , b 1 and c 1 be the direction ratios of the first plane. So,

a 1 =2 b 1 =1 c 1 =3

Let a 2 , b 2 and c 2 be the direction ratios of the second plane. So,

a 2 =1 b 2 =−2 c 2 =0

If the planes are perpendicular, then,

a 1 a 2 + b 1 b 2 + c 1 c 2 =0 2( 1 )+1( −2 )+3( 0 )=0 2−2−0=0 0=0

Therefore, the planes are perpendicular to each other.

(c)

The equations of the two given planes are,

2x−2y+4z+5=0

And

3x−3y+6z−1=0

Let a 1 , b 1 and c 1 be the direction ratios of the first plane. So,

a 1 =2 b 1 =−2 c 1 =4

Let a 2 , b 2 and c 2 be the direction ratios of the second plane. So,

a 2 =3 b 2 =−3 c 2 =6

If the planes are perpendicular, then,

a 1 a 2 + b 1 b 2 + c 1 c 2 =0 2( 3 )+( −2 )( −3 )+4( 6 )=0 6+6+24=0 36≠0

This shows that the planes are not perpendicular.

If the planes are parallel, then,

a 1 a 2 = b 1 b 2 = c 1 c 2 2 3 = −2 −3 = 4 6 2 3 = 2 3 = 2 3

Therefore, the planes are parallel to each other.

(d)

The equations of the two given planes are

2x−y+3z−1=0

And

2x−y+3z+3=0

Let a 1 , b 1 and c 1 be the direction ratios of the first plane. So,

a 1 =2 b 1 =−1 c 1 =3

Let a 2 , b 2 and c 2 be the direction ratios of the second plane. So,

a 2 =2 b 2 =−1 c 2 =3

If the planes are perpendicular, then,

a 1 a 2 + b 1 b 2 + c 1 c 2 =0 2( 2 )+( −1 )( −1 )+3( 3 )=0 4+1+9=0 14≠0

This shows that the planes are not perpendicular.

If the planes are parallel, then,

a 1 a 2 = b 1 b 2 = c 1 c 2 2 2 = −1 −1 = 3 3 1=1=1

Therefore, the planes are parallel to each other.

(e)

The equations of the two given planes are

4x+8y+z−8=0

And

y+z−4=0

Let a 1 , b 1 and c 1 be the direction ratios of the first plane. So,

a 1 =4 b 1 =8 c 1 =1

Let a 2 , b 2 and c 2 be the direction ratios of the second plane. So,

a 2 =0 b 2 =1 c 2 =1

If the planes are perpendicular, then,

a 1 a 2 + b 1 b 2 + c 1 c 2 =0 4( 0 )+8( 1 )+1( 1 )=0 8+1=0 9≠0

This shows that the planes are not perpendicular.

If the planes are parallel, then,

a 1 a 2 = b 1 b 2 = c 1 c 2 4 0 = 8 1 = 1 1 ∞≠8≠1

This shows that the planes are not parallel.

The angle between the planes is,

θ= cos −1 | a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 ⋅ a 2 2 + b 2 2 + c 2 2 | = cos −1 | 4( 0 )+8( 1 )+1( 1 ) 4 2 + 8 2 + 1 2 × 1 2 + 1 2 | = cos −1 | 8+1 16+64+1 × 1+1 | = cos −1 | 9 81 × 2 |

Solve further,

θ= cos −1 | 9 9× 2 | = cos −1 ( 1 2 ) =45°

Therefore, the angle between the planes is 45°.