specialist mathematics vectors and geometry week 3
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Specialist Mathematics
Vectors and GeometryWeek 3
Example 13 (Ex 4F)ABC angle of measure theFind
8cm
3cm
2cmA
B
C
Solution 13ABC angle of measure theFind
8cm
3cm
2cm
B
C
A
Vector Product
22
11
22
11
22
11
22
11
22
11
22
11
222
111
222111
,,aor
athen
,, and ,,
yxyx
xzxz
zyzy
b
yxyx
kxzxz
jzyzy
i
zyxzyxkji
b
zyxbzyxaIf
Example 14 (Ex 4G1)
abbacbacbaIf
(c) . (b) (a) find3,1,2 and 1,0,2 ,3,2,1
Solution 14
abbacbacbaIf
(c) . (b) (a) find3,1,2 and 1,0,2 ,3,2,1
Direction of a x b. and both lar toperpendicu is Note baba
ba
ab
ab
ab-ba Note
Length of a x b
sinbaba
ba
00sin Note o
baba toparallel is 0 therefore
Properties of a x b• a x b is a vector perpendicular to both a and b.
• a x a = 0
• a x b = - b x a
• a x (b + c) = a x b + a x c
• (a + b) x (c + d) = a x c + a x d + b x c + b x d
Scalar Triple Product
33
221
33
221
33
221
333
222
111
333222111
then
,, and ,, ,,,
yxyx
zxzxz
yzyzy
x
zyxzyxzyx
cba
zyxczyxbzyxaIf
Example 15 (Ex 4G1) cba
cba
find
1,0,2 and 1,2,3 ,1,1,2 If
Solution 15 cba
cba
find
1,0,2 and 1,2,3 ,1,1,2 If
Example 16 (Ex 4G1) bab 2a3simplify and Expand
Solution 16 bab 2a3simplify and Expand
Example 17 (Ex 4G1)
.3,0,2 and 1,3,1both lar toperpendicu vectorsall Find
ba
Solution 17
.3,0,2 and 1,3,1both lar toperpendicu vectorsall Find
ba
Area of a Triangle
a
b
ba21 Area
Area of a Parallelogram
a
b
ba Area
Volume of a Parallelepiped
a
bc
333
222
111
Volume
zyxzyxzyx
cba
Volume of a Tetrahedron
a
bc
333
222
111
61
61Volume
zyxzyxzyx
cba
Example 18 (Ex 4H)
130C and 143B ,302Afor ABC triangle theof area theFind
,,-,-,-,,
Solution 18
130C and 143B ,302Afor ABC triangle theof area theFind
,,-,-,-,,
Example 19 (Ex 4H)
1,1,0D and 130C,143B ,302Afor ABCDon tetrahedr theof volume theFind
,,-,-,-,,
Solution 19
1,1,0D and 130C,143B ,302Afor ABCDon tetrahedr theof volume theFind
,,-,-,-,,
Coplanar Points
0ADACAB
ifcoplaner be willD and CB,A, Thereforezero. is points theup makingon tetrahedrthe
of volume theifcoplaner are spacein pointsFour
Example 20 (Ex 4H)
coplaner? 5,-3,-1D and1,5,2C,0,2,3B ,4,2,1A points theAre
Solution 20
coplaner? 5,-3,-1D and1,5,2C,0,2,3B ,4,2,1A points theAre
This weeks work
• Text Pages 146 to 156.• Exercise 4F Q 12 – 21• Exercise 4G1 Q 1 – 14• Exercise 4G2 Q 2 – 5• Exercise 4H Q 1 - 10
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