TOPIC 8: VECTORS 1 (BASIC PROPERTIES + SCALAR/VECTOR PRODUCTS) Flashcards
Representation of a vector
AB (arrow on top)
x (wavy line at bottom)
Representation of the magnitude of a vector
|AB (arrow on top)|
|x (wavy line at bottom)|
2 Types of Vector
Free vector
* Direction vector
Localised vector
* Position vector
* Displacement vector
3 Laws of Scalar Multiplication
Commutative
Associative
Distributive
[Formula] Unit Vector
â = a/|a|
Determine if vector is parallel
If one can be expressed as a scalar multiple of the other
Collinearity Theorem
- Both lines parallel
- Have a common point
[Formula] Ratio Theorem
refer to notes page 11
Conventional Unit Vectors along x,y,z axes
x axis: i
y axis: j
z axis: k
2 Main Basic Concepts of 3D Vectors
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)
[Formula] Scalar Product / Dot Product
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)
[Property] Is Scalar Product commutative?
Yes
a.b = b.a
[Property] Is Scalar Product associative?
Yes
a.(λb) = (λa).b
[Property] Is Scalar Product distributive?
Yes
a.(b.c) = (a.b).c
[Property] 4 Main Properties of Scalar Product
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
Find angle between 2 vectors
a.b = |a||b|cosθ
cosθ = (a.b) / (|a||b|) = (unit vector of A).(unit vector of B)
[Formula] Length of projection of a onto b
OF = |a.b|/|b|
also equals to
OF = |a . (unit vector of b)|
[Formula] Projection Vector of a onto b
Vector OF (with arrow on top) =
OF.(unitvectorb) OR (a.unitvectorb)(unitvectorb)
Proof:
Vector OF = kb
k = OF
Vector OF = (a.unitvectorb)(unitvectorb)
[Projection] How to know if projection vector and a/b are in the same/opposite direction?
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
[Formula] Vector Product / Cross Product
a x b = (|a||b|sinθ)(unitvector of n)
unitvector of n is the unit vector perpendicular to both a and b
[Property] Is Vector Product commutative?
No. a x b = - (b x a)
[Property] Is Vector Product associative?
Yes.
k(a x b) = (ka) x b = a x (kb)
[Property] Is Vector Product distributive?
Yes.
a x (b + c) = a x b + a x c
[Property] 3 Main Properties of Vector Product
a x b = (|a||b|sinθ)(unit vector of n)
- |a x b| = |a||b|sinθ
- IF a and b are parallel,
a x b = 0 - IF a and b are perpendicular,
a x b = |a||b|
Vector Product of unit vectors along the axes
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
How to verify if vector product is done correctly?
(a x b).a = (a x b).b = 0
How to find area of triangle using vector product?
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|
How to find area of parallelogram using vector product?
Think of it as 2 triangles
Area of parallelogram ABCD = |vector AB x vector AD|
*2 vectors need to be side by side
Give a geometrical intepretation of |p x q|
The area of a parallelogram with sides p and q
Perpendicular Distance from OA to OF, which is AF
AF = |a x b|/|b|
also equals to
OF = |a x (unit vector of b)|
