Vectors 3 - Dot Product PDF

# Lecture 3 Dot Product - **Q. Should the product of two vectors be scalar or vector?** - **A. Both!** ## 1.8 Scalar Product - Scalar product is defined as: - **a ⋅ b = |a||b|cos θ** - Scalar or dot product ### Geometrically: - b - Projection of b onto a = bcosθ = (a ⋅ b) / |a| = â (a...

# Lecture 3 Dot Product - **Q. Should the product of two vectors be scalar or vector?** - **A. Both!** ## 1.8 Scalar Product - Scalar product is defined as: - **a ⋅ b = |a||b|cos θ** - Scalar or dot product ### Geometrically: - b - Projection of b onto a = bcosθ = (a ⋅ b) / |a| = â (a ⋅ b) / |a|² ## 1.9 Dot Product Rules - **Rule 2i:** a ⋅ b = b ⋅ a (commutative - MAMMA MIA!) - **Rule 3:** a ⋅ b = |a||b|cosθ = |a||b|cos(-θ) = b ⋅ a - **Rule 3i:** - a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) (distributive) - From figure: b + c = b + c - a ⋅ (b + c) = a ⋅ b + a ⋅ c (as above) - **Rule 3ii:** - If a ⊥ b, θ = π/2, cosθ = 0 and - a ⋅ b = 0 (orthogonality) - Hence for basis i, j, k: - i ⋅ i = j ⋅ j = k ⋅ k = 1 but i ⋅ j = j ⋅ k = i ⋅ k = 0 - Take a = axi + ayj + azk b = bx i + by j + bz k - a ⋅ b = (axi + ayj + azk) ⋅ (bxi + byj + bzk) - = axbx + ayby + azbz - In general in R^n, a ⋅ b = Σ aibi (summation for each row) - a ⋅ b = (ax ay az) ⋅ (bx by bz) - **Rule 3iv:** - a ⋅ a = Σ ai ai = Σ a²i = a₁² + a₂² + ... + an² = |a|² (length square) - If you like a² = (a ⋅ a) ## 1.10 Finding Angles - Find angle between a = (2 1) b = (3 4) - cos θ = a ⋅ b / |a||b| = (2 1) ⋅ (3 4) / √(2² + 1²) √(3² + 4²) = 6 + 4 / √5 √25 = √24 / 5√5 - θ = cos⁻¹(√24 / 5√5) = 79.7°. ## 1.11 Applications of dot product ### 1.11.1 Physical uses - e.g. Work done W = F ⋅ d ⇒ power = F ⋅ v (constant force) - Elect. dipole in E field, UE = - p ⋅ E (e-dipole moment) - mag. dipole in B field, UB = -M ⋅ B (mag. dipole moment) ### 1.11.2 Flux - flow of particles nV through area. - define A = Aâ - "volume" flowing through in Δt: - ΔV = Vi Δt A = VCOSO A Δt = V ⋅ ΔA Δt = (V ⋅ A) Δt - ⇒ number flowing thru' ΔN = n(V ⋅ A) Δt - ⇒ and rate of flow = FLUX = ΔN/Δt = nV ⋅ A (flow vector) - can be applied to any vector, - e.g. flux of E-field, ΦE = E ⋅ Δ - flux of B-field, ΦB = B ⋅ Δ ## 1.11.3 Collisions - what is Φ? - mom. conserved P = p1 + p2 - ⇒ m1v1 = m1v1' + m1v2' - energy cons. 1/2m1 v1² = 1/2m1 v1'² + 1/2 m1 v2'² - u² = u1² + u2'² - but u² = (u1 + u2) = (u1 + u2) ⋅ (u1 + u2) = |u1|² + |u1 ⋅ u2| + 2u1 ⋅ u2 + |u2|² - ⇒ u1² + 2u1 ⋅ u2 + u2² = u1² + u2² - ⇒ u1 ⋅ u2 = 0 - cos Φ = 0, Φ = 90°, u1 ⊥ u2!