The Dot Product. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics

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1 The Dot Product MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2015

2 Dot Product The dot product is an operation performed on two vectors (the inputs) which produces a scalar (the output). Definition The dot product of two vectors a = a 1, a 2, a 3 and b = b 1, b 2, b 3 in V 3 is defined by a b = a 1, a 2, a 3 b 1, b 2, b 3 = a 1 b 1 + a 2 b 2 + a 3 b 3. Likewise the dot product of two vectors a = a 1, a 2 and b = b 1, b 2 in V 2 is defined by a b = a 1, a 2 b 1, b 2 = a 1 b 1 + a 2 b 2. The dot product is sometimes called the inner product.

3 Examples Compute the dot products of the following vectors. 1. a = 2, 1, 3 and b = 1, 4, 7 2. a = 2i + 3j + 4k and b = 3i + 4j 5k

4 Examples Compute the dot products of the following vectors. 1. a = 2, 1, 3 and b = 1, 4, 7 a b = (2)( 1) + (1)(4) + (3)(7) = a = 2i + 3j + 4k and b = 3i + 4j 5k a b = ( 2)(3) + (3)(4) + (4)( 5) = 14

5 Properties of the Dot Product Theorem For vectors a, b, and c and any scalar d, the following hold: 1. a b = b a (commutativity) 2. a (b + c) = a b + a c (distributive law) 3. (da) b = d(a b) = a (db) 4. 0 a = 0 5. a a = a 2.

6 Geometry of the Dot Product (1 of 2) For two nonzero vectors the angle between the vectors is the smaller angle between the two vectors when their initial points are the same.

7 Geometry of the Dot Product (1 of 2) For two nonzero vectors the angle between the vectors is the smaller angle between the two vectors when their initial points are the same. Theorem Let θ be the angle between nonzero vectors a and b. Then, a b = a b cos θ.

8 Geometry of the Dot Product (2 of 2) Proof. Suppose a and b are not parallel. a b Θ a According to the Law of Cosines: b a b 2 = a 2 + b 2 2 a b cos θ.

9 Example Find the angle between the vectors a = 3, 4, 1 and b = 2, 5, 7. Express the angle in both radian and degree measure.

10 Example Find the angle between the vectors a = 3, 4, 1 and b = 2, 5, 7. Express the angle in both radian and degree measure. a b = a b cos θ (3)(2) + ( 4)(5) + ( 1)(7) = cos θ cos θ = ( θ = cos 1 21 )

11 Orthogonality Two vectors are called orthogonal (or perpendicular) if the angle between them is θ = π 2. The zero vector is orthogonal to every vector.

12 Orthogonality Two vectors are called orthogonal (or perpendicular) if the angle between them is θ = π 2. The zero vector is orthogonal to every vector. Corollary Two vectors a and b are orthogonal if and only if a b = 0.

13 Examples 1. Determine if a = 1, 3, 5 and b = 2, 3, 10 are orthogonal.

14 Examples 1. Determine if a = 1, 3, 5 and b = 2, 3, 10 are orthogonal. a b = (1)(2)+(3)(3)+(5)(10) = 61 0 = not orthogonal

15 Examples 1. Determine if a = 1, 3, 5 and b = 2, 3, 10 are orthogonal. a b = (1)(2)+(3)(3)+(5)(10) = 61 0 = not orthogonal 2. Determine the value of z 2 which makes a = 2, 3, 14 and b = 4, 2, z 2 orthogonal.

16 Examples 1. Determine if a = 1, 3, 5 and b = 2, 3, 10 are orthogonal. a b = (1)(2)+(3)(3)+(5)(10) = 61 0 = not orthogonal 2. Determine the value of z 2 which makes a = 2, 3, 14 and b = 4, 2, z 2 orthogonal. 0 = a b = (2)(4) + (3)(2) + 14z 2 = z 2 z 2 = 1

17 Cauchy-Schwartz Inequality Theorem (Cauchy-Schwartz Inequality) For any vectors a and b, a b a b.

18 Cauchy-Schwartz Inequality Theorem (Cauchy-Schwartz Inequality) For any vectors a and b, a b a b. Proof. If a 0 and b 0 then a b = a b cos θ a b.

19 Triangle Inequality Theorem (Triangle Inequality) For any vectors a and b, a + b a + b.

20 Triangle Inequality Theorem (Triangle Inequality) For any vectors a and b, a + b a + b. Proof. a + b 2 = (a + b) (a + b) = a a + a b + b a + b b = a a + 2a b + b b = a 2 + 2a b + b 2 a a b + b 2 = ( a + b ) 2

21 Components a Θ a cosθ b The length of the shadow cast by a parallel to b is the component of a along b.

22 Components (Another Perspective) a a cosθ Θ b The length of the shadow cast by a parallel to b is the component of a along b.

23 Definition of Components Definition The scalar quantity called the component of a along b is a cos θ. It is denoted comp b a = a cos θ = a b b cos θ = 1 a b a b cos θ = b b

24 Definition of Components Definition The scalar quantity called the component of a along b is a cos θ. It is denoted comp b a = a cos θ = a b b cos θ = 1 a b a b cos θ = b b Remark: the component of a along b is the dot product of a with a unit vector in the same direction as b.

25 Projections Definition The projection of a onto b is the vector parallel to b whose magnitude is the component of a along b. proj b a = (comp b a) b b = (a b)b b 2

26 Projections Definition The projection of a onto b is the vector parallel to b whose magnitude is the component of a along b. proj b a = (comp b a) b (a b)b = b b 2 Remark: comp b a is a scalar, but proj b a is a vector.

27 Examples (1 of 2) Let a = 6i 3j + 2k and b = 2i + 5j k. 1. Find comp b a 2. Find proj b a

28 Examples (1 of 2) Let a = 6i 3j + 2k and b = 2i + 5j k. 1. Find comp b a comp b a = a b b = Find proj b a

29 Examples (1 of 2) Let a = 6i 3j + 2k and b = 2i + 5j k. 1. Find comp b a comp b a = a b b = Find proj b a proj b a = (a b)b b 2 = 29 2, 5, 1 = , 29 6, 29 30

30 Examples (2 of 2) Let a = 6i 3j + 2k and b = 2i + 5j k. Find proj a b

31 Examples (2 of 2) Let a = 6i 3j + 2k and b = 2i + 5j k. Find proj a b proj a b = (b a)a a 2 = 29 6, 3, 2 = , 87 49, 58 49

32 Example A beam in a skyscraper is installed in the direction of 10, 1, 5. It is subject to a force parallel to the direction 13, 180, 6. Find the component and projection of the force along the beam.

33 Example A beam in a skyscraper is installed in the direction of 10, 1, 5. It is subject to a force parallel to the direction 13, 180, 6. Find the component and projection of the force along the beam. Let b = 10, 1, 5 and F = 13, 180, 6, then comp b F = b F 2 b = proj b F = (comp b F) b b = , 40 63,

34 Homework Read Section Exercises: 1 63 odd.

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