Progress to date has used only basic vector operations: addition, subtraction, scalar multiplication, and division.
Question arises: what is the utility in multiplying vectors?
Functions as a useful operation with more than one method to form a 'product' of two vectors.
Overview of vector operations so far:
Scalar Operations:
Addition/Subtraction: π + π = π, π - π = π
Integration/Differentiation: ππ/ππ‘ or β«πππ‘
Multiplication/Division: π Γ π = π, π Γ· π = π
Vector Operations:
Addition/Subtraction: πβ1 + πβ2 = πβ3, πβ3 β πβ2 = πβ1
Integration/Differentiation: ππβ/ππ‘ or β«πβππ‘
Scalar multiplication/division: ππβ1 = πβ2, πβ1 = πβ2/π
Defined for two non-collinear vectors πβ1 and πβ2 in a 3D space:
These vectors form a plane, not necessarily perpendicular.
Cross product gives a vector πβ3 that is perpendicular to the plane formed by πβ1 and πβ2.
Magnitude equals the area of the parallelogram defined by πβ1 and πβ2.
Important notes on collinearity:
Collinear vectors do not define a plane, making the cross product zero in such cases.
Area of the parallelogram can be calculated using:
π3 = |π1||π2| sin(π)
Valid only when π β€ 180Β°
Indicates zero cross product for collinear vectors (π = 0 or Ο).
Determined using the right-hand rule:
Point fingers of the right hand along πβ1 and curl towards πβ2.
Thumb direction indicates the cross product direction.
Ordering matters; reversing vectors in cross product reverses the resultant vector's direction.
Vectors defined by components:
πβ1 = π1,π₯πΜ + π1,π¦πΜ + π1,π§πΜ
πβ2 = π2,π₯πΜ + π2,π¦πΜ + π2,π§πΜ
In a right-handed coordinate system:
πΜ Γ πΜ = πΜ, πΜ Γ πΜ = πΜ, πΜ Γ πΜ = π.
Distributing terms to find the expression:
πβ1 Γ πβ2 = (π1,π¦π2,π§ β π1,π§π2,π¦)πΜ + (π1,π§π2,π₯ β π1,π₯π2,π§)πΜ + (π1,π₯π2,π¦ β π1,π¦π2,π₯)πΜ
Application primarily uses magnitude from area formulas and the right-hand rule.
Cross product yields a vector; dot product produces a scalar (size only, no direction).
Scalar defined by coordinate independence. Changing coordinates does not change the scalar description.
The dot product's essence:
Magnitude Calculation: π = β(ππ₯Β² + ππ¦Β² + ππ§Β²)
Dot Product: πβ1βπβ2 = π1,π₯π2,π₯ + π1,π¦π2,π¦ + π1,π§π2,π§
Applications in work and energy calculations; itβs commutative: πβ1βπβ2 = πβ2βπβ1.
When magnitudes |π1| and |π2| are known along with angle π:
πβ1βπβ2 = |π1||π2|cos(π).
Dot product sign:
Positive: acute angle (0 β€ π < Ο/2)
Negative: obtuse angle (Ο/2 < π β€ Ο)
Zero: angle = Ο/2.
Vectors: πβ1 = 1πΜ + 2πΜ + 0πΜ, πβ2 = β2πΜ + 1πΜ + 0πΜ
Compute using components:
πβ1 Γ πβ2 = (0 + 0 + 5πΜ)
Utilizing area and right-hand rule:
|πβ1 Γ πβ2| = 5, direction along +πΜ.
Vectors: πβ1 = 1πΜ + 2πΜ + 0πΜ, πβ2 = β2πΜ + 1πΜ + 0πΜ
Compute using components:
πβ1 β πβ2 = 0.
Utilizing magnitudes and angle:
πβ1 β πβ2 = 0 (angle = 90Β°).
Understanding both cross and dot products is crucial in physics for analyzing motion, forces, torque, and energy.