6-4 Vectors and dot products Chapter 6 6-4 Vectors and dot products

Objectives Find the dot product of two vectors and use the properties Find the angle between two vectors Write vectors as the sum of two vectors components Use vectors to find the work done by a force

Dot product of two vectors Dot Product is a third vector operation. This vector operation yields a scalar (a single number) not another vector. The dot product can be positive, zero or negative. Definition of dot product

Example This is called the dot product. Notice the answer is just a number NOT a vector.

Example #1: Finding dot products Find each dot product. A) <4, 5>●<2, 3> sol: 23 B) <2, -1>●<1, 2> sol:0 C) <0, 3>●<4, -2> sol:-6 D) <6, 3>●<2, -4> sol: 0 E) (5i + j)●(3i – j) sol:14

Properties of Dot products Let u, v, and w be vectors in the plane or in space and let c be a scalar. u●v = v●u 0●v = 0 u●(v + w) = u●v + u●w v●v = ||v||2 c(u●v) = cu●v = u●cv

Example#2 Using properties of dot products Let u=<-1,3>, v=<2,-4> and w=<1,-2>. Use vectors and their properties to find the indicated quantity A. (u.v)w sol: <-14,28> B.u.2v sol: -28 C.||u|| sol:√10

Check it out! Let u = <1, 2>, v = <-2, 4> and w = <-1, -2>. Find the dot product. A) (u●v)w B) u●2v

Check it out ! Given the vectors u = 8i + 8j and v = —10i + 11j find the following. A. B.

Angle between two vectors Angle between two vectors (θ is the smallest non- negative angle between the two vectors)

Example Find the angle between

solution

solution

Check it out! Find the angle between u = <4, 3> and v = <3, 5>. Solution: 22.2 degrees

Definitions of orthogonal vectors The vectors u and v are orthogonal if u●v = 0. Orthogonal = Perpendicular = Meeting at 90°

Example: Orthogonal vectors Are the vectors u = <2, -3> and v = <6, 4> orthogonal?

Example Determine if the pair of vectors is orthogonal

Definition of Vector Components Let u and v be nonzero vectors. u = w1 + w2 and w1 · w2 = 0 Also, w1 is a scalar of v The vector w1 is the projection of u onto v, So w1 = proj v u w 2 = u – w 1

Decomposing of a Vector Using Vector Components Find the projection of u into v. Then write u as the sum of two orthogonal vectors Sol:

Solution

Work The work W done by a constant force F in moving an object from A to B is defined as This means the force is in some direction given by the vector F but the line of motion of the object is along a vector from A to B

Definition of work Work is force times distance. If Force is a constant and not at an angle If Force is at an angle

Example To close a barn’s sliding door, a person pulls on a rope with a constant force of 50 lbs. at a constant angle of 60 degrees. Find the work done in moving the door 12 feet to its closed position. Sol: 300 lbs

Example Find the work done by a force of 50 kilograms acting in the direction 3i + j in moving an object 20 metres from (0, 0) to (20, 0).

solution Let's find a unit vector in the direction 3i + j Our force vector is in this direction but has a magnitude of 50 so we'll multiply our unit vector by 50.

solution

Student guided practice Do7,9,11,23,31,43 and 57 in your book page 440 and 441

Homework Do 8,10,12,14,24,32,44 and 69 in your book page 440 and 441

Closure Today we learned about vectors and work Next class we are going to learn about trigonometric form of a complex number