USED TO PROVED COMPOUND ANGLE FORMULA OF SINE
Sin (A + B) = sin A cos B + sin B cos A
Consider the vector diagram below
Pg. drawing
= (cos A) 
+ (sin A) 
+ o
= (cos A) 
+ (sin A) 
+ o
Hence
= 
sin (A + B)
But
= 
= i 
+ k
= i (o) – j (o) + k (-cos A sin B – sin A cos B)
= -k [sin A cos B + cos A sin B]
= 
2
= 
2
= sin A cos B + cos A sin B
Also
= 
= 
2
2 = 
2 B
= 1
Therefore
Sin A cos B + cos A sin B = (1) (1) sin (A+B)
Sin (A + B) = Sin A cos B + cos A sin B
Proved
USED TO DETERMINE/ TO PROVE COMPOUND AND FORMULA OF SINE
Sin (A – B) = sin A cos B – Cos A sin B
– Consider the vector diagram below
Pg.6 drawing
= (cos A) 
+ (sin A) 
+ o
= (cos B) 
+ (sin B) 
+ o
= 
= 
sin (A – B)
but
= 
= i 
– j 
+ k 
= 
2
= 
2
= sin A cos B – cos A sin B
Also
= 
+ 0
= 
= 
= 1
= 
= 
= 
= 1
Therefore
Sin A cos B – cosA sin B = (1) (1) sin (A – B)
Sin (A – B) = sin A cos B – cos A sin B
Proved
USED TO FIND THE AREA OF THE TRIANGLE
– Consider the triangle ABC below
Pg. 7 drawing
Area (A) = ½ x base (b) x height (h)
A = ½ bh ……i
Also
Sin 
= 
Sin 
= 
h = 
Sin 
….ii
And
b = 
…iii
Substitute …ii and …iii into 1 as follows
A = ½ 
sin 
A = ½ 
USED TO FIND THE AREA OF PARALLELOGRAM
– Consider the parallelogram below
Pg. 7 drawing
Area = length (l) x height (h)
= A = Lh …i
L = 
…ii
Also
Sin Q = 
h = 
Sin Q….ii
Substitute ii and iii into —1 as follows
Area (A) = 
sin Q
A = 
Generally
Area (A) = 
= 
= 
= 
Where
= 
– 
= 
– 
= 
– 
= 
– 
USED TO FIND THE VOLUME OF PARALLELOPIPED
– Consider the diagram below
Pg. 7 drawing
– Suppose P Q R and S are the vertices of the parallelepiped, hence the volume (v) of the parallelepiped is given by:
Volume (v) = base area (A) x height (h)
V = 
x 
V = 
V = 
V = 
Again, for the sides with position vectors
, 
and 
Volume (v) = 
x 
= 
= 
USED TO FIND THE VOLUME OF A TETRAHEDRON
– Consider the tetrahedron with vertices P, Q, R and S
Pg. 8 drawing
Volume (v) = 1/3 x base area x altitude
V = 1/3 x ½ 
x 
V = 1/6 
x 
V = 
Therefore
Volume = 1/6 
USED
TO FIND THE VECTOR PERPENDICULAR TO THE PLANE
– Consider the diagram below
Pg. 8 drawing
Where
= is the vector perpendicular or normal to the plane
= 
x 
Question
26. Find the area of the triangle ABC whose vertices are A (2, 1, 1) B (3, 2, 1) and C (-2, -4, -1)
27. The position vector of the points A, B and C are (2, 4, 3), (6, 3, -4) and (7, 5, -5) respectively. Find the angle between 
and 
and hence the area of the triangle ABC
28. Find the area of the parallelogram whose vertices P, Q and R are (2, 1, 1), (3, 2, 1) (2, 4, 1)
29. Find the volume of the parallelepiped the edges are A (1, 0, 2) B (2, -1, 3) C (4, 1, 3) and D (1, -1, 1)
30. Find the volume of the tetrahedron whose sides are 
= 2
+ 
, 
= 
– 3
` + 
, and 
– 
+ 2