1. Find the Dot Product of Two Vectors 2. Find the Angle Between Two Vectors

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1 Objectives kˆz 1. Find the Dot Product of Two Vectors 2. Find the Angle Between Two Vectors t < 0 r 0 t > 0 ĵy 3. Determine if Two Vectors Are Parallel 4. Determine if Two Vectors Are Orthogonal 5. Decompose a Vector into Two Orthogonal Vectors îx 6. Compute Work 1 Kidoguchi, Kenneth

2 The Dot Product So far we have added two vectors and multiplied a vector by a scalar. The question arises: Is it possible to multiply two vectors so that their product is a useful quantity? An example of a situation in physics and engineering where we need to combine two vectors occurs in calculating the work done by a force. We defined the work done by a constant force F in moving an object through a distance d as W = Fd, but this applies only when the force is directed along the line of motion of the object.

3 Work and the Dot Product Suppose, however, that the constant force is a vector F = PR pointing in some other direction, as in Figure 1. Figure 1 If the force moves the object from P to Q, then the displacement vector is D = PQ. So here we have two vectors: the force F and the displacement D.

4 Work and the Dot Product We use this concept to define the dot product of two vectors even when they don t represent force or displacement. Definition: The dot product of two nonzero vectors a and b is the number: a b = a b cos θ where q is the angle between a and b when the vectors are placed tail-totail and 0 θ π. If either a or b is the zero vector 0 or if θ = π 2, then a b = 0. This product is often called the dot product because of the dot in the notation a b. The result of computing a b is not a vector. It is a real number, that is, a scalar. For this reason, the dot product is sometimes called the scalar product.

5 Example 1 Sample Computation If the vectors a and b have lengths 4 and 6, and the angle between them is /3, find a b.

6 Expanded Definition Given two vectors: u = iu 1 + ju 2 + ku 3 and v = iv 1 + jv 2 + kv 3 the "dot product" of these two vectors is defined by the relation: u v u v cosq u v u v u v where: u u1 u2 u3 u u, v v v v v v, and 0 < q <, is the angle between the two vectors when the vectors are placed tail-to-tail. u q 1 2 The dot product of u and v is read u dot v. 3 v 26 May, Kidoguchi, Kenneth

7 Sample Computation Vector in 2D Given: u 3, 4 and w 2,5 find the dot product of u onto w and find the angle between the two vectors. 8 Kidoguchi, Kenneth

8 Sample Computation Vector in 2D Given: u 3, 4 and w 2,5 find the dot product of u onto w and find the angle between the two vectors. u w u and w u w cos q u w cos q q arccos, , 0 q q cos q q ĵ î 26 May, Kidoguchi, Kenneth

9 Sample Computation Vector in 3D Given: u 3, 0, 4 and w 1, 1, 4 find the dot product of u onto w and find the angle between the two vectors.

10 Sample Computation Vector in 3D Given: u 3, 0, 4 and w 1, 1, 4 find the dot product of onto and find the angle between the two vectors. u w 3( 1) 01 4( 4) u and w ( 1) ( 1) ( 4) 3 2 u w u w u w cos q 15 2 cos q 19 cos 19 q q arccos, q May, Kidoguchi, Kenneth

11 Vector Algebra Samples Given: a 0,2,1, b 3,5,4, c 1,6,0, evaluate: 1. a b 2. a b c 3. c c a a 26 May, Kidoguchi, Kenneth

12 The Law of Cosines Revisited c b q a 26 May, Kidoguchi, Kenneth

13 Parallel Vectors Two vectors v and w are said to be parallel if there is a nonzero scalar a such that v aw. In this case, q between v and w is 0 or. Example: Determine if the vectors v 4iˆ 8 ˆj and w 2iˆ 4 ˆj are parallel. 16 Kidoguchi, Kenneth

14 Orthogonal Vectors If the angle q between two nonzero vectors are said to be orthogonal. and Two vectors v and w are said to be orthogonal if and only if: v w 0 Example: v w is /2, the vectors Determine if the vectors v 1,1 and w 1,1 are orthogonal. 17 Kidoguchi, Kenneth

15 Dot Product Interpretation The "projection of u onto v ". The "projection of v onto u ". u q v u v u cos(q) u v v q v cos(q) u v u 26 May, Kidoguchi, Kenneth

16 Dot Product Vector Projections Find the vector projection of v 1, 3 onto w 1,1. Decompose v into two vectors v and v where v is parallel to w and v is orthogonal to w Kidoguchi, Kenneth

17 Dot Product Vector Projections Find the vector projection of v 1, 3 onto w 1,1. Decompose v into two vectors v and v where v is parallel to w and v is orthogonal to w vw v w v w cos q v cos q w vw w 11 2 vw v cosq scalar projection of v onto w w Kidoguchi, Kenneth

18 Dot Product Vector Projections v1 v cos q scalar projection of v onto w w v1 v w v cosq vector projection of v onto w w 2 v w w 2 w 2 2 1, 1 w 2 2 w w v v v v w Kidoguchi, Kenneth

19 Dot Product Work Work = F d, is the dot product of the exerted (assumed constant) Force, F, and displacement, d. A crate is pushed up a frictionless ramp that makes an angle a with the floor. If the ramp is d metres long and the force exerted on the crate is F Newtons vertically upward, then the work done by the force is: d Work Fd F F d cos q j i a q 26 May, Kidoguchi, Kenneth Fdcos / 2 a Fdsin Notes: Work is a scalar and the SI unit for work is a Joule. Force is a vector and the SI unit for force is a Newton. a

20 Dot Product Scalar Work and Vector Force A worker pushes a crate up a frictionless ramp that makes an angle a = 30º with the floor. If the ramp is 10 metres long and the magnitude of the total force exerted on the crate is 100 Newtons which is equal to the weight of the crate, then: a) Compute the work done by the force. b) Compute the component of the force d exerted by the worker to do this work. Work Fd F F d cosq j i a q F d 26 May, Kidoguchi, Kenneth Fdcos / 2 a 10010sin 500 Joules / 6 The force exerted by the worker, F d is the projection of F onto d. Hence: F d = Fcos(q) = 50 Nt.

21 Dot Product Scalar Work and Vector Force A worker pushes a crate up a frictionless ramp that makes an angle a = 45º with the floor. If the ramp is 10/ 2 metres long and the magnitude of the total force exerted on the crate is 100 Newtons which is equal to the weight of the crate, then: a) Compute the work done by the force. b) Compute the component of the force exerted by the worker to do this work. F d Work Fd q Fd F d cos q Fdcos / 2 a / 2 sin 500 Joules / 4 j i a The force exerted by the worker, F d is the projection of F onto d. Hence: F d = Fcos(q) = 50 2 Nt. 26 May, Kidoguchi, Kenneth

22 Dot Product Scalar Work and Vector Force Dagmar is pulling a wagon with a force of 40 pounds. Present the analysis to find the work done in moving the wagon 200 feet if the handle makes an angle of 30 with the ground.

23 Vector Application Dagmar runs a 100-metre dash on a track which is in the direction of vector v = 4, 12. The wind velocity is w = 5, 2 km/hr. Rules require that the wind speed in the direction of the race must not exceed 5 km/hr. Assume that the wind vector remains constant. Present the analysis to determine if this race will be disqualified.