Dot Product. Academic Resource Center

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Lambert Day
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1 Dot Product Academic Resource Center

2 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

3 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

4 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.

5 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

6 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) = a b = Given a = 2j + 7k and b = -7i + 4j - k a b = (0)(-7) + (2)(4) + (7)(-1) = (-7) a b = 1

b 2 a b = (-4)(2) + (-5)(-9) = -8 + 45 a b = 37 2.

7 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)

8 Dot Product If the angle between the two vectors a and b is θ, then a b = a b cos θ or θ =

9 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)

10 Example Find the angle between the two vectors: Given a = <-1, -3> and b = <3, -3> a b = (-1)(3) + (-3)(-3) = = 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

+ 9 = 6 a = (1+9) = (10) b = (9+9) = (18) = 3 (2) θ =

11 Direction Angles Given a = <a 1, a 2, a 3 > cos α = cos β = cos γ =

12 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

13 Projections Scalar projection of b onto a: comp a b = Vector projection of b onto a: proj a b = ( ) =

14 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) = = 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>

-3 + 9 = 6 a = (1+9) = (10) b = (9+9) = (18) = 3 (2) comp a b

15 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

The handle of the cart is pulled at an angle of 60 above the horizontal.

16 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) = W = -21 J

Q(1,4,-7) Solution: d = <1 5, 4 3, -7 (-4)> = <-4, 1, -3> W = F d

17 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

two other coordinate systems Given a vector A = A x a x + A y a y + A z a z we

18 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

19 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

20 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

21 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

22 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

23 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

24 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

25 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

26 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)

27 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

28 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)

29 References Calculus Stewart 6 th Edition, Section 13.3 The Dot Product

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v, 1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It