Specialist Mathematics Vectors and Geometry Week 3.
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Specialist Mathematics
Vectors and GeometryWeek 3
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Example 13 (Ex 4F)ABC angle of measure theFind
8cm
3cm
2cmA
B
C
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Solution 13ABC angle of measure theFind
8cm
3cm
2cm
B
C
A
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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
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Example 14 (Ex 4G1)
abbacbacbaIf
(c) . (b) (a) find3,1,2 and 1,0,2 ,3,2,1
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Solution 14
abbacbacbaIf
(c) . (b) (a) find3,1,2 and 1,0,2 ,3,2,1
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Direction of a x b. and both lar toperpendicu is Note baba
ba
ab
ab
ab-ba Note
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Length of a x b
sinbaba
ba
00sin Note o
baba toparallel is 0 therefore
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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
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Scalar Triple Product
33
221
33
221
33
221
333
222
111
333222111
then
,, and ,, ,,,
yxyx
zxzxz
yzyzy
x
zyxzyxzyx
cba
zyxczyxbzyxaIf
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Example 15 (Ex 4G1) cba
cba
find
1,0,2 and 1,2,3 ,1,1,2 If
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Solution 15 cba
cba
find
1,0,2 and 1,2,3 ,1,1,2 If
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Example 16 (Ex 4G1) bab 2a3simplify and Expand
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Solution 16 bab 2a3simplify and Expand
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Example 17 (Ex 4G1)
.3,0,2 and 1,3,1both lar toperpendicu vectorsall Find
ba
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Solution 17
.3,0,2 and 1,3,1both lar toperpendicu vectorsall Find
ba
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Area of a Triangle
a
b
ba21 Area
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Area of a Parallelogram
a
b
ba Area
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Volume of a Parallelepiped
a
bc
333
222
111
Volume
zyxzyxzyx
cba
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Volume of a Tetrahedron
a
bc
333
222
111
61
61Volume
zyxzyxzyx
cba
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Example 18 (Ex 4H)
130C and 143B ,302Afor ABC triangle theof area theFind
,,-,-,-,,
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Solution 18
130C and 143B ,302Afor ABC triangle theof area theFind
,,-,-,-,,
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Example 19 (Ex 4H)
1,1,0D and 130C,143B ,302Afor ABCDon tetrahedr theof volume theFind
,,-,-,-,,
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Solution 19
1,1,0D and 130C,143B ,302Afor ABCDon tetrahedr theof volume theFind
,,-,-,-,,
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Coplanar Points
0ADACAB
ifcoplaner be willD and CB,A, Thereforezero. is points theup makingon tetrahedrthe
of volume theifcoplaner are spacein pointsFour
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Example 20 (Ex 4H)
coplaner? 5,-3,-1D and1,5,2C,0,2,3B ,4,2,1A points theAre
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Solution 20
coplaner? 5,-3,-1D and1,5,2C,0,2,3B ,4,2,1A points theAre
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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