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