TOPIC 8: VECTORS 1 (BASIC PROPERTIES + SCALAR/VECTOR PRODUCTS)

Question 1

Representation of a vector

Answer 1

AB (arrow on top)
x (wavy line at bottom)

Question 2

Representation of the magnitude of a vector

Answer 2

|AB (arrow on top)|
|x (wavy line at bottom)|

Question 3

2 Types of Vector

Answer 3

Free vector
* Direction vector

Localised vector
* Position vector
* Displacement vector

Question 4

3 Laws of Scalar Multiplication

Answer 4

Commutative
Associative
Distributive

Question 5

[Formula] Unit Vector

Answer 5

â = a/|a|

Question 6

Determine if vector is parallel

Answer 6

If one can be expressed as a scalar multiple of the other

Question 7

Collinearity Theorem

Answer 7

1. Both lines parallel
2. Have a common point

Question 8

[Formula] Ratio Theorem

Answer 8

refer to notes page 11

Question 9

Conventional Unit Vectors along x,y,z axes

Answer 9

x axis: i
y axis: j
z axis: k

Question 10

2 Main Basic Concepts of 3D Vectors

Answer 10

r = xi + yj + zk = x(1 0 0) + (0 1 0) + z (0 0 1)

*x/y/z is the magnitude, i/j/k is the direction (unit vectors)
*the 001 is presented in vertical form

|r| = √(x² + y² + z²)

Question 11

[Formula] Scalar Product / Dot Product

Answer 11

a.b = |a||b|cosθ

*θ is the angle between a and b ONLY WHEN they both converge / diverge
* 0° ≤ θ ≤ 180°

OR

a = (a₁a₂a₃) ; b = (b₁b₂b₃)
a.b = (a₁b₁ + a₂b₂ + a₃b₃)

Question 12

[Property] Is Scalar Product commutative?

Answer 12

Yes
a.b = b.a

Question 13

[Property] Is Scalar Product associative?

Answer 13

Yes
a.(λb) = (λa).b

Question 14

[Property] Is Scalar Product distributive?

Answer 14

Yes
a.(b.c) = (a.b).c

Question 15

[Property] 4 Main Properties of Scalar Product

Answer 15

a.a = |a|²

|a.b| = |a||b| (ONLY IF a and b are PARALLEL)

a.b = 0 (ONLY IF a and b are PERPENDICULAR)

If θ is acute, a.b > 0
If θ is obtuse, a.b < 0

Question 16

Find angle between 2 vectors

Answer 16

a.b = |a||b|cosθ
cosθ = (a.b) / (|a||b|) = (unit vector of A).(unit vector of B)

Question 17

[Formula] Length of projection of a onto b

Answer 17

OF = |a.b|/|b|

also equals to

OF = |a . (unit vector of b)|

Question 18

[Formula] Projection Vector of a onto b

Answer 18

Vector OF (with arrow on top) =
OF.(unitvectorb) OR (a.unitvectorb)(unitvectorb)

Proof:
Vector OF = kb
k = OF
Vector OF = (a.unitvectorb)(unitvectorb)

Question 19

[Projection] How to know if projection vector and a/b are in the same/opposite direction?

Answer 19

Length of projection = a.b/|b|

If a.b > 0, angle between a and b is acute, vector OF and b are in the same direction

If a.b < 0, angle between a and b is obtuse, vector OF and b are in the opposite direction

Question 20

[Formula] Vector Product / Cross Product

Answer 20

a x b = (|a||b|sinθ)(unitvector of n)

unitvector of n is the unit vector perpendicular to both a and b

Question 21

[Property] Is Vector Product commutative?

Answer 21

No. a x b = - (b x a)

Question 22

[Property] Is Vector Product associative?

Answer 22

Yes.
k(a x b) = (ka) x b = a x (kb)

Question 23

[Property] Is Vector Product distributive?

Answer 23

Yes.
a x (b + c) = a x b + a x c

Question 24

[Property] 3 Main Properties of Vector Product

Answer 24

a x b = (|a||b|sinθ)(unit vector of n)

1. |a x b| = |a||b|sinθ
2. IF a and b are parallel,
   a x b = 0
3. IF a and b are perpendicular,
   a x b = |a||b|

Question 25

Vector Product of unit vectors along the axes

Answer 25

i x i = j x j = k x k = 0

i x j = k ; j x k = i ; k x i = j

j x i = -k ; k x j = -i ; i x k = -j

Question 26

How to verify if vector product is done correctly?

Answer 26

(a x b).a = (a x b).b = 0

Question 27

How to find area of triangle using vector product?

Answer 27

Area of triangle ABC = 1/2 |vector AB x vector BC|

*AB / BC can be any 2 sides

Proof:
Area of triangle = 1/2|a||b|sinθ
a x b = |a||b|sinθ
So, area of traingle = 1/2|a x b|

Question 28

How to find area of parallelogram using vector product?

Answer 28

Think of it as 2 triangles

Area of parallelogram ABCD = |vector AB x vector AD|

*2 vectors need to be side by side

Question 29

Give a geometrical intepretation of |p x q|

Answer 29

The area of a parallelogram with sides p and q

Question 30

Perpendicular Distance from OA to OF, which is AF

Answer 30

AF = |a x b|/|b|

also equals to

OF = |a x (unit vector of b)|