contents 12.1 angle between two straight lines 12.2 angle between a straight line and a plane 12.3...
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Contents
12.1 Angle between Two Straight Lines
12.2 Angle between a Straight Line and a Plane
12.3 Angle between Two Planes
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12.1 Angle between Two Straight Lines
The angle between two intersecting straight lines l1 and l2 is given by the acute angle between the two straight lines as shown in Fig. 12.6. If two straight lines on a plane do not intersect, they are parallel.
Fig 12.6
In three dimensions, when two lines do intersect, we can simply consider the plane containing these intersecting lines, as shown in Fig. 12.8.
In three dimensions, the angle between two intersecting straight lines is the acute angle between the straight lines lying on the same plane,
Fig 12.8
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12.1 Angle between Two Straight Lines
Fig.12.10 shows a right pyramid VABCD with a square base. The length of each slant edge is 10 cm and each side of the base is 8 cm long. VN is the height of the right pyramid.
Example 12.1
(b) Find the angle between the line VB and the line VD.
(Give the answers correct to 3 significant figures.)
(a) Find the angle between the line VB and the line VC.
Fig 12.10
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12.1 Angle between Two Straight Lines
BVC
BVC
BVCVCVBVCVBBC
cos20010010064
cos)10)(10(210108
cos))((2222
222
Solution:
(a) Consider a triangle with sides VB and VC, in this case, VBC.
)nt figures significao (correct t
BVC
BVC
32.47
1563.47200
136cos
Since VBC is an isosceles triangle, students may use another method by drawing the altitude bisecting VBC and using simple trigonometric ratio to find the required angle.
Fig 12.10
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(b) Consider a triangle with sides VB and VD, in this case, VBD.
24
128
88 222
DN
DB
DB
4499.3410
24sin
DVNDV
DNDVN
12.1 Angle between Two Straight Lines
Solution:
)nt figures significao (correct t
DVNBVD
39.68
8998.68
2
Fig 12.10
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When a line cuts a plane at a point P like a javelin hitting a lawn as shown in Fig. 12.31, the angle between the line and the plane is defined as the angle between PA and PB, where PB is the projection of PA on the plane.
In three dimensions, the angle between a straight line and a plane is the acute angle between the straight line and its projection on the plane.
12.2 Angle between a Straight Line and a Plane
PA, PB and the vertical line AB form a right-angled triangle perpendicular to the plane. Therefore, it will be easier to find the angle between a straight line and a plane if we identify a right-angled triangle involved.
Fig 12.31
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12.3 Angle between Two Planes
In three dimensions, the angle between two planes is the acute angle between two perpendiculars on the respective planes to the intersection of the two planes.
The angle between two planes is defined as the acute angle between the perpendiculars to the intersecting lines of the two planes.
Fig. 12.54
As shown in Fig. 12.54, the intersection of the two planes is PQ. AP and CQ lie on Plane I. BP and DQ lie on Plane II. AP PQ and BP PQ. Therefore, APB() is the angle between two planes.