06. TO PROVE THAT AN INSCRIBED ANGLE SUBTENDING A SEMI – CIRCLE IS A RIGHT ANGLE
– Consider the vector diagram below.
Pg. 2 drawing
To prove that
< S
R = 900
. 
= 0
-b + 
+ – 
= 0
– 
+ 
= 
= 
+ 
+ 
+ –
= 0
+ 
= 
= 
– 
Hence
. 
= (
+ 
) (
– 
)
. 
= 
2 – (
2
. 
= 
2 – 
2
. 
= 
2 – 
2
But
= 
= radius, r
. 
= 0
Proved
QUESTION
17. Find the projection of 
+ 2
– 3
onto 
+ 2
+ 2
18. Find the vector projection of 
onto
. If 
= 2
+2
+ 
and 
= 3
+ 
+ 2
19. Find the work done of the force of (2i + 3i + k) n s pulling a load (3i + j k) m
20. Find the work done of the force of (2i + 3j + k) N is pulling a load a distance of 2m in the direction of 2m in the direction of
= 3
+ 2
+ 2
21. Find a vector which has magnitude of 14 in the direction of 2
+ 3
+ 
CROSS (VECTOR) PRODUCT (X or 
)
x 
= 
sin Ø. 
Where
– is the unit vector perpendicular to both vector 
and 
= 
x 
= 
Hence
x 
= 
sin Ø. 
1 = 
= 
sin Ø
Therefore
= 
sin Ø
OR
= 
sin Ø
Where
Ø – is the angle between the vector 
and 
Again
Suppose the vector
Hence
+ 
= 
+ 
= 
– j 
+ 
Note
i) If you cross two vectors, the product is also the vector.
ii) Cross (vector) product uses the knowledge of determinant of 3 x 3 matrix.
iii) From the definition.
= 
sin 
Individual
= (1, 0, 0)
j = (0, 1, 0)
k = (0, 0, 1)
= (1) (1) (0)
= 0
= 0
Hence
i x i = j x 
= 
x 
= 0
iv) From the definition
x 
= k
Generally
Consider the component vector
For anticlockwise (+ve)
Pg. 4 drawing
i) 
x 
= 
ii) 
x 
= 
iii) 
x 
= 
For clockwise (-ve)
i) 
x 
= 
ii) 
x 
= – 
iii) 
x 
= – 
THEOREM
From the definition
– 
= 0
Is the condition for collinear (parallel) vectors
a) 
≠ 
= – 
Questions
22. If 
= 2
+ 6
+ 3
and 
= 
+ 2
+ 2
. Find the angle between 
and 
23. Determine a unit vector perpendicular to 
= 2
– 6
– 3
and 
= 4
+ 3
– 
24. If 
= 2
+ j + 2k and 
= 3
+ 2
+ 
Find 
â‹€
Box product
-This involves both cross and dot product
Suppose
. 
x 
then start with cross (x) followed by DOT (.)
. 
x 
=
. 
x
)
-This is sometimes called scalar triple product
Note
-If scalar triple product (box product) of three vectors
.
and 
= 0
-Then the vector 
, 
and 
are said to be COMPLANAR
Question
25. If 
= 2
+ 
+ 2
= 2
+ 
and
= 3
+ 2
+ k
Find
a) 
. 
x c
b) 
x
. 
APPLICATION OF CROSS PRODUCT
USED TO PROVE SINE RULE
– Consider the diagram below
+ 
+ – 
= 0
+ 
= 
Cross by 
on both sides of
x 
+ 
x 
= 
x 
0 + 
x 
= 
x 
x 
+ 
x 
Crossing by 
on both sides of eqn …. 1 above
+ 
= 
x 
+ 
+ 
= 
x 
x 
+ 
= 
x 
x 
= 
x c
– (
x 
= 
x 
x 
= – 
x 
)
x 
= 
x 
Equation i and ii as follows
x 
= 
x 
= 
x 
Sin
.
= 
sin
.
= 
sin
.
Sin
= 
Sin
= 
sin 
Dividing the whole eqn by
= 
=
= 
= 
Sine rule