Math Textbook: 6.4 Vectors and Dot Products

Finding Dot Products: (0,3)*(4,-2)=?

(0,3)*(4,-2) =0(4)+3(-2) =0-6 =-6

Finding Dot Products: (2,-1)*(1,2)=?

(2,-1)*(1,2) =2(1)+(-1)2 =2-2 =0

Finding Dot Products: (4,5)*(2,3)=?

(4,5)*(2,3) =4(2)+5(3) =8+15 =23

‖projPQF‖ is equal to what?

(cosθ)‖F‖

Finding a Force: A 200 pound cart sits on a ramp inclined at 30°, as shown in Figure 6.28. What force is required to keep the cart from rolling down the ramp?

1. Because the force due to gravity is vertical and downward, you can represent the gravitational force by the vector F=-200J 2. To find the force required to keep the cart from rolling down the ramp, project F onto a unit vector v in the direction of the ramp, as follows v=(cos30°)i+(sin30°)j =√3/2i+1/2j 3. So, the projection of F onto v is w₁=projvF=(F*v/‖v‖²)v =(F*v)v =(-200)(1/2)v =-100(√3/2i+1/2j) 4. The magnitude of this force is 100, so a force of 100 pounds is required to keep the cart from rolling down the ramp.

What is the equation relating F to its components w₁ and w₂?

F=w₁+w₂

What does w₂ represent?

w₂ represents the force that the tires must withstand against the ramp

How many possible orientations of two vectors are there?

Five possible orientations.

Using Properties of Dot Products: Let u=(-1,3), v=(2,-4) and w=(1,-2). Find ‖u‖

1. Begin by finding the dot product of u and u u*u =(-1,3)*(-1,3) =-1(-1)+3(3) =1+9 =10 2. Because ‖u‖²=u*u=10, it follows that ‖u‖=√u*u =√10

Using Properties of Dot Products: Let u=(-1,3), v=(2,-4) and w=(1,-2). Find (u*v)w

1. Begin by finding the dot product of u and v u*v =(-1,3)*(2,-4) =-1(2)+3(-4) =-2-12 =-14 2. (u*v)w =-14(1,-2) =(-14,28)

Using Properties of Dot Products: Let u=(-1,3), v=(2,-4) and w=(1,-2). Find u*2v

1. Begin by finding the dot product of u and v u*v =(-1,3)*(2,-4) =-1(2)+3(-4) =-2-12 =-14 2. u*2v =2(u*v) =2(-14) =-28

Determining Orthogonal Vectors: Are the vectors u=(2,-3) and v=(6,4) orthogonal?

1. Find the dot product of the two vectors u*v=(2,-3)*(6,4) =2(6)+4(-3) =12-12 =0 2. Because the dot product is 0, the two vectors are orthogonal.

The work W done by a constant force F as its point of application moves along the vector→PQ is given by what 2 formulas?

1. Projection Form W=‖projPQF‖‖PQ‖ 2. Dot Product Form W=F*PQ

Decomposing a Vector into Components: Find the projection of u=(3,-5) onto v=(6,2). Then write u as the sum of two orthogonal vectors, one of which is projvu.

1. The projection of u onto v is w₁=projvu =projvu=(u*v/‖v‖²)v =(8/40)(6,2) =(6/5, 2/5) 2. The other component, w₂ is w₂=u-w₁ =(3,-5)-(6/5, 2/5) =(9/5, -27/5) 3. So, u=w₁+w₂ =(6/5, 2/5)+(9/5, -27/5) =(3,-5)

Finding the Work Done: To close a barn's sliding door, a person pulls on a rope with a constant force of 50 pounds at a constant angle of 60°, as shown in Figure 6.31. Find the work done in moving the door 12 feet to its closed position.

1. Using a projection, you can calculate the work as follows W=‖projPQF‖‖PQ‖ =(cos60°)‖F‖‖PQ‖ =(1/2)(50)(12) =300 foot-pounds 2. So, the work done is 300 foot-pounds

Finding the Angle Between Two Vectors: Find the angle θ between u=(4,3) and v=(3,5)

1. cosθ=u*v/‖u‖‖v‖ =(4,3)*(3,5)/‖(4,3)‖‖(3,5)‖ =27/5√34 2. This implies that the angle between the two vectors is θ=arccos(27/5√34) =22.2°

Let u, v, and w be vectors in the plane or in space and let c be a scalar. Finish the 5 Properties of the Dot Product 1. u*v=? 2. 0*v=? 3. u*(v+w)=? 4. v*v=? 5. c(u*v)=?

1. u*v=v*u 2. 0*v=0 3. u*(v+w)=u*v+u*w 4. v*v=‖v‖² 5. c(u*v)=cu*v or u*cv

From this form, you can see that _____, ______ will always have _____.

From this form, you can see that because ‖u‖ and ‖v‖ are always positive, u*v and cosθ will always have the same sign.

What is the third kind of vector operation and what does it yield?

The Dot Product. This product yields a scalar, rather than a vector.

What is the angle between two nonzero vectors?

The angle θ, 0≤θ≤π, between their respective standard position vectors

Consider a boat on an inclined ramp. The force ___ due to gravity pulls the boat ___ the ramp and _____ the ramp. These two orthogonal forces, w₁ and w₂, are _____ of ___.

The force F due to gravity pulls the boat DOWN the ramp and AGAINST the ramp. These two orthogonal forces, w₁ and w₂, are VECTOR COMPONENTS of F.

What does the negative of component w₁ represent?

The negative of component w₁ represents the force needed to keep the boat from rolling down the ramp

How can the angle between two nonzero vectors be found?

This angle can be found using the dot product

Let u and v be nonzero vectors such that u=w₁+w₂ where w₁ and w₂ are orthogonal and w₁ is parallel to (or a scalar multiple of) v. The vectors w₁ and w₂ are called _____. The vector w₁ is the ______ and is denoted by ______. The vector w₂ is given by _____.

Vector components of u. The vector w₁ is the projection of u onto v and is denoted by w₁=proj₁u. (the 1 is actually a v) The vector w₂ is given by w₂=u-w₁

The work W done by a constant force F acting along the line of motion of an object is given by...?

W=(magnitude of force)(distance) W=‖F‖‖PQ‖

If the constant force F is not directed along the line of motion, then the work W done by the force is given by...?

W=‖projPQF‖‖PQ‖ =(cosθ)‖F‖‖PQ‖ =F*PQ

The vectors u and v are orthogonal (perpendicular) when....?

When u*v=0

If θ is the angle between two nonzero vectors u and v, then...?

cosθ=u*v/‖u‖‖v‖

Let u and v be nonzero vectors. The projection of u onto v is given by.....?

proj₀u=(u*v/‖v‖²)v (the 0 is actually a "v")

The dot product of u=(u₁,u₂) and v=(v₁,v₂) is...?

u*v=u₁v₁+u₂v₂

Rewriting the expression for the angle between two vectors in the form _____ produces an alternative way to _____

u*v=‖u‖‖v‖cosθ produces an alternative way to calculate the dot product.

Find the projection of u onto v when u=w₁+w₂

u=w₁+w₂ 1. w₁ is a scalar multiple of v u=cv+w₂ 2. Take dot product of each side with v u*v=(cv+w₂)*v u*v=cv*v+w₂*v u*v=c‖v‖²+0 3. So, c=u*v/‖v‖² 4. And, w₁=projvu =cv =(u*v/‖v‖²)v