Vectors
2.3 Scalar (Dot) Product
Scalar (dot) product definition and component form, angle between vectors, projection, cross product determinant form, right-hand rule, scalar and vector triple products.
2.3 Scalar (Dot) Product
Definition 2.15: a·b = |a||b|cos θ
Component form: a·b = a₁b₁ + a₂b₂ + a₃b₃
Component form: a·b = a₁b₁ + a₂b₂ + a₃b₃
Unit vector dot products:
i·j = j·k = i·k = 0 (perpendicular)
i·i = j·j = k·k = 1 (parallel, same)
i·j = j·k = i·k = 0 (perpendicular)
i·i = j·j = k·k = 1 (parallel, same)
Example 2.15: a = 2i+3j+5k, b = 4i+j+6k
(a) a·b = 2×4 + 3×1 + 5×6 = 8+3+30 = 41
(b) |a| = 4+9+25 = 38
(c) |b| = 16+1+36 = 53
(d) cos θ = 4138·53 = 412014 ≈ 0.914
(e) θ = arccos(0.914) ≈ 23.9°
Projection of a Vector
Projection of a onto b: projb(a) = (a·b|b|²) b
Scalar projection: compb(a) = a·b|b|
Scalar projection: compb(a) = a·b|b|
Orthogonality Condition
a ⊥ b ⟺ a·b = 0 (when both are nonzero vectors)
2.4 Vector (Cross) Product
Definition 2.19: a×b = (ab sin θ)n̂
where n̂ is the unit vector perpendicular to both a and b, direction by right-hand rule.
Magnitude |a×b| = ab|sin θ| = area of parallelogram formed by a and b.
where n̂ is the unit vector perpendicular to both a and b, direction by right-hand rule.
Magnitude |a×b| = ab|sin θ| = area of parallelogram formed by a and b.
Determinant Form (most useful)
a×b = |i, j, k; a₁, a₂, a₃; b₁, b₂, b₃|
= (a₂b₃−a₃b₂)i − (a₁b₃−a₃b₁)j + (a₁b₂−a₂b₁)k
= (a₂b₃−a₃b₂)i − (a₁b₃−a₃b₁)j + (a₁b₂−a₂b₁)k
Unit vector cross products (right-hand rule):
i×j=k, j×k=i, k×i=j
j×i=−k, k×j=−i, i×k=−j
i×i=j×j=k×k=0
i×j=k, j×k=i, k×i=j
j×i=−k, k×j=−i, i×k=−j
i×i=j×j=k×k=0
Cross product is NOT commutative: a×b = −(b×a)
Scalar Triple Product
a·(b×c) = |a₁,a₂,a₃; b₁,b₂,b₃; c₁,c₂,c₃| = Volume of parallelepiped
Applications
- Work: W = F·d = |F||d|cos θ
- Moment/Torque: M = r×F
- Angular velocity: v = ω×r