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: 𝑎 + 𝑏 = 𝑐, 𝑐 - 𝑏 = 𝑎
Integration/Differentiation: 𝑑𝑎/𝑑𝑡 or ∫𝑎𝑑𝑡
Multiplication/Division: 𝑎 × 𝑏 = 𝑐, 𝑐 ÷ 𝑏 = 𝑎
Vector Operations:
Addition/Subtraction: 𝑉⃗1 + 𝑉⃗2 = 𝑉⃗3, 𝑉⃗3 − 𝑉⃗2 = 𝑉⃗1
Integration/Differentiation: 𝑑𝑉⃗/𝑑𝑡 or ∫𝑉⃗𝑑𝑡
Scalar multiplication/division: 𝑘𝑉⃗1 = 𝑉⃗2, 𝑉⃗1 = 𝑉⃗2/𝑘
Defined for two non-collinear vectors 𝑉⃗1 and 𝑉⃗2 in a 3D space:
These vectors form a plane, not necessarily perpendicular.
Cross product gives a vector 𝑉⃗3 that is perpendicular to the plane formed by 𝑉⃗1 and 𝑉⃗2.
Magnitude equals the area of the parallelogram defined by 𝑉⃗1 and 𝑉⃗2.
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:
𝑉3 = |𝑉1||𝑉2| sin(𝜃)
Valid only when 𝜃 ≤ 180°
Indicates zero cross product for collinear vectors (𝜃 = 0 or π).
Determined using the right-hand rule:
Point fingers of the right hand along 𝑉⃗1 and curl towards 𝑉⃗2.
Thumb direction indicates the cross product direction.
Ordering matters; reversing vectors in cross product reverses the resultant vector's direction.
Vectors defined by components:
𝑉⃗1 = 𝑉1,𝑥𝑖̂ + 𝑉1,𝑦𝑗̂ + 𝑉1,𝑧𝑘̂
𝑉⃗2 = 𝑉2,𝑥𝑖̂ + 𝑉2,𝑦𝑗̂ + 𝑉2,𝑧𝑘̂
In a right-handed coordinate system:
𝑖̂ × 𝑗̂ = 𝑘̂, 𝑗̂ × 𝑘̂ = 𝑖̂, 𝑘̂ × 𝑖̂ = 𝑗.
Distributing terms to find the expression:
𝑉⃗1 × 𝑉⃗2 = (𝑉1,𝑦𝑉2,𝑧 − 𝑉1,𝑧𝑉2,𝑦)𝑖̂ + (𝑉1,𝑧𝑉2,𝑥 − 𝑉1,𝑥𝑉2,𝑧)𝑗̂ + (𝑉1,𝑥𝑉2,𝑦 − 𝑉1,𝑦𝑉2,𝑥)𝑘̂
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: 𝑉 = √(𝑉𝑥² + 𝑉𝑦² + 𝑉𝑧²)
Dot Product: 𝑉⃗1∙𝑉⃗2 = 𝑉1,𝑥𝑉2,𝑥 + 𝑉1,𝑦𝑉2,𝑦 + 𝑉1,𝑧𝑉2,𝑧
Applications in work and energy calculations; it’s commutative: 𝑉⃗1∙𝑉⃗2 = 𝑉⃗2∙𝑉⃗1.
When magnitudes |𝑉1| and |𝑉2| are known along with angle 𝜃:
𝑉⃗1∙𝑉⃗2 = |𝑉1||𝑉2|cos(𝜃).
Dot product sign:
Positive: acute angle (0 ≤ 𝜃 < π/2)
Negative: obtuse angle (π/2 < 𝜃 ≤ π)
Zero: angle = π/2.
Vectors: 𝑉⃗1 = 1𝑖̂ + 2𝑗̂ + 0𝑘̂, 𝑉⃗2 = −2𝑖̂ + 1𝑗̂ + 0𝑘̂
Compute using components:
𝑉⃗1 × 𝑉⃗2 = (0 + 0 + 5𝑘̂)
Utilizing area and right-hand rule:
|𝑉⃗1 × 𝑉⃗2| = 5, direction along +𝑘̂.
Vectors: 𝑉⃗1 = 1𝑖̂ + 2𝑗̂ + 0𝑘̂, 𝑉⃗2 = −2𝑖̂ + 1𝑗̂ + 0𝑘̂
Compute using components:
𝑉⃗1 ∙ 𝑉⃗2 = 0.
Utilizing magnitudes and angle:
𝑉⃗1 ∙ 𝑉⃗2 = 0 (angle = 90°).
Understanding both cross and dot products is crucial in physics for analyzing motion, forces, torque, and energy.