Dot Product Academic Resource Center
In This Presentation We will give a definition Look at properties See the relationship in projections Look at vectors in different coordinate systems Do example problems
Dot Product Definition: If a = <a 1, a 2 > and b = <b 1, b 2 >, then the dot product of a and b is number a b given by a b = a 1 b 1 + a 2 b 2 Likewise with 3 dimensions, Given a = <a 1, a 2, a 3 > and b = <b 1, b 2, b 3 > a b = a 1 b 1 + a 2 b 2 + a 3 b 3
Dot Product The result of a dot product is not a vector, it is a real number and is sometimes called the scalar product or the inner product.
Orthogonal Vectors Two vectors a and b are orthogonal (perpendicular) if and only if a b = 0 Example: The vectors i, j, and k that correspond to the x, y, and z components are all orthogonal to each other
Examples Find a b: 1. Given a = <-4, -5> and b = <2, -9> a b = a 1 b 1 + a 2 b 2 a b = (-4)(2) + (-5)(-9) = -8 + 45 a b = 37 2. Given a = 2j + 7k and b = -7i + 4j - k a b = (0)(-7) + (2)(4) + (7)(-1) = 0 + 8 + (-7) a b = 1
Dot Product Properties of the dot product 1. a a = a 2 2. a b = b a 3. a (b + c) = a b + a c 4. (ca) b = c(a b) = a (cb) 5. 0 a = 0 (Note that 0 (bolded) is the zero vector)
Dot Product If the angle between the two vectors a and b is θ, then a b = a b cos θ or θ =
Examples Find a b: 1. Given a = 8, b = 4 and θ = 60 a b = a b cos θ = (8)(4)cos(60 ) = 12(1/2) a b = 6 2. Given a = 3, b = 2 and θ = π/4 a b = (3)(2)cos(π/4) = 6( (2)/2) a b = 3 (2)
Example Find the angle between the two vectors: Given a = <-1, -3> and b = <3, -3> a b = (-1)(3) + (-3)(-3) = -3 + 9 = 6 a = (1+9) = (10) b = (9+9) = (18) = 3 (2) θ = 6/( (10)*3 (2)) = 6/(6 (5)) = 1/ (5) θ = 1/ (5) = (5)/5 rad
Direction Angles Given a = <a 1, a 2, a 3 > cos α = cos β = cos γ =
Example Find the direction angles of the vector a: Given a = 3i j + 4k a = (9+1+16) = (26) cos α = 3/ (26) α = cos -1 (3/ (26)) = 0.94 rad cos β = -1/ (26) β = cos -1 (-1/ (26)) = 1.8 rad cos γ = 4/ (26) γ = cos -1 (4/ (26)) = 0.67 rad
Projections Scalar projection of b onto a: comp a b = Vector projection of b onto a: proj a b = ( ) =
Example Find the scalar and vector projection of b onto a: Given a = <-1, -3> and b = <3, -3> a b = (-1)(3) + (-3)(-3) = -3 + 9 = 6 a = (1+9) = (10) b = (9+9) = (18) = 3 (2) comp a b = a b/ a = 6/ (10) proj a b = a b/ a 2 *a = 6/10*a = <-3/5, -9/5>
Application Example 1 Problem: A cart is pulled a distance of 50m along a horizontal path by a constant force of 25 N. The handle of the cart is pulled at an angle of 60 above the horizontal. Find the work done by the force. Solution: F and d are force and displacement vectors W = F d = F d cosθ = (25)(50)cos(60 ) W = 625 J
Application Example 2 Problem: Given a constant vector field F = 7i + 3j k find the work done from point P(5,3,-4) to the point Q(1,4,-7) Solution: d = <1 5, 4 3, -7 (-4)> = <-4, 1, -3> W = F d = <7,3,-1> <-4,1,-3> = (7)(-4) + (3)(1) + (-1)(-3) = -28 + 4 +3 W = -21 J
Vectors Given a vector in any coordinate system, (rectangular, cylindrical, or spherical) it is possible to obtain the corresponding vector in either of the two other coordinate systems Given a vector A = A x a x + A y a y + A z a z we can obtain A = A ρ a ρ + A Φ a Φ + A z a z and/or A = A r a r + A Φ a Φ + A θ a θ
Rectangular Coordinate System Define A = A x a x + A y a y + A z a z as a rectangular vector where each component is a function of x, y, and z
Cylindrical Coordinate System Define A = A ρ a ρ + A Φ a Φ + A z a z as a rectangular vector where each component is a function of ρ, Φ, and z
Spherical Coordinate System Define A = A r a r + A Φ a Φ + A θ a θ as a rectangular vector where each component is a function of r, Φ, and θ
Rectangular to Cylindrical Dot products of unit vectors in cylindrical and rectangular coordinate systems a x a x a x a x cos Φ -sin Φ 0 x = ρ cosφ y = ρ sinφ z = z a y sin Φ cos Φ 0 a z 0 0 1
Rectangular to Spherical Dot products of unit vectors in spherical and rectangular coordinate systems x = r sinθ cosφ y = r sinθ sinφ z = r cosθ a r A θ a Φ a x sin θ cos Φ cos θ cos Φ -sin Φ a y sin θ sin Φ cos θ sin Φ cos Φ a z cos θ -sin θ 0
Conversion Given a rectangular vector A = A x a x + A y a y + A z a z, we want to find the vector in cylindrical coordinates A = A ρ a ρ + A Φ a Φ + A z a z To find any desired component of a vector, we take the dot product of the vector and a unit vector in the desired direction. A ρ = A a ρ and A Φ = A a Φ
Example 1 Express the vector F = 4a x -2a y +8a z in cylindrical coordinates: F ρ = F a ρ = 4(a x a ρ ) 2(a y a ρ ) + 8(a z a ρ ) = 4(cosΦ) 2(sinΦ) + 8(0) = 4cosΦ 2sinΦ F Φ = F a Φ = 4(a x a Φ ) 2(a y a Φ ) + 8(a z a Φ ) = 4(-sinΦ) 2(cosΦ) + 8(0) = -4sinΦ 2cosΦ F z = 8 F = (4cosΦ 2sinΦ)a ρ + (-4sinΦ 2cosΦ)a Φ + 8a z
Example 1 (cont.) Evaluate F given ρ = 2.5, Φ = 0.7, z = 1.5: F = (4cosΦ 2sinΦ)a ρ + (-4sinΦ 2cosΦ)a Φ + 8a z F ρ = 4cosΦ 2sinΦ = 4cos(0.7) 2sin(0.7) = 1.77 F Φ = -4sinΦ 2cosΦ = -4sin(0.7) 2cos(0.7) = -4.1 F z = 8 F = 1.77a ρ 4.1a Φ + 8a z
Example 2 Given a vector field E = xa x + ya y + za z, convert to cylindrical and spherical coordinates: E ρ = E a ρ = x(a x a ρ ) + y(a y a ρ ) + z(a z a ρ ) = ρcosφ(cosφ) ρsinφ(sinφ) + z(0) = ρcos 2 Φ + ρsin 2 Φ = ρ(cos 2 Φ + sin 2 Φ) = ρ E Φ = E a Φ = x(a x a Φ ) + y(a y a Φ ) + z(a z a Φ ) E z = z = ρcosφ(-sinφ) + ρsinφ(cosφ) + z(0) = -ρcosφsinφ + ρcosφsinφ = 0 E = ρa ρ + za z (cylindrical)
Example 2 (cont.) Express E in spherical coordinates: E r = E a r = x(a x a r ) + y(a y a r ) + z(a z a r ) = x(sinθcosφ) + y(sinθsinφ) + z(cosθ) = rsin 2 θcos 2 Φ + rsin 2 θsin 2 Φ + rcos 2 θ = rsin 2 θ(cos 2 Φ + sin 2 Φ) + rcos 2 θ = rsin 2 θ + rcos 2 θ = r(sin 2 θ + cos 2 θ) = r E θ = E a θ = x(a x a θ ) + y(a y a θ ) + z(a z a θ ) = x(cosθcosφ) + y(cosθsinφ) + z(-sinθ) = rsinθcosθcos 2 Φ + rcosθsinθsin 2 Φ rsinθcosθ = rsinθcosθ(cos 2 Φ + sin 2 Φ) rsinθcosθ = rsinθcosθ rsinθcosθ = 0
Example 2 (cont.) E Φ = E a Φ = x(a x a Φ ) + y(a y a Φ ) + z(a z a Φ ) = x(-sinφ)+ y(cosφ) = -rsinθsinφcosφ + rsinθsinφcosφ = 0 E = r a r (spherical)
References Calculus Stewart 6 th Edition, Section 13.3 The Dot Product