Lines & Planes In 3d

MATHEMATICS SPM LINES AND PLANES IN 3-DIMENSIONS “Open and Close” Method to identify i. Angles between Lines and Planes (close method) ii. Angles between Two Planes (open method) MOHD NAZAN BIN KAMARUL ZAMAN SMK. KOTA KLIAS, BEAUFORT

Example 1 The following diagram shows a cuboid KLMNPQRS Identify the angle between the line LP and the plane KLMN

S

R

Q

P

M

N

K

L

Solution i. Draw the line LP and the plane KLMN

P N

K

M

L

ii. ‘Close’ the line LP and the plane KLMN together

P M

N

P

K

L

The line LP down straight to line KL, so the straight line we use is line KL

iii. ‘Open’ the line LP (180º)

M

N

p

K

L

P

Identify three points when we joint together become a straight line. The straight line is KLP or PLK , so the angle between the line LP and the plane KLMN is ∠ KLP or ∠ PLK

Answer:

S

R

Q

P

M

N

K

L

The answer is ∠ KLP or ∠ PLK

Example 1 The following diagram shows a right prism ABCHED with a horizontal plane ABCD. It is a uniform prism and its cross section is an isosceles triangle of sides 4 cm. The thickness of the prism, EA = 4 cm. H

E

A

D

C

B

Identify the angle between the plane ABH and the plane ABCD.

Solution i. Draw the plane ABH and the plane ABCD

H

DD

A

C

B

iii. ‘Open’ the plane ABH (180º) H

D

A

H

C

D

B

Therefore the angle between the two planes is ∠ DAH @ ∠ HAD