1. 6.4 Vectors and Dot Products Day 1 2015

2. Precalculus HWQ
Find the magnitude and direction of the resultant of the following vectors:
u has magnitude 300 at an angle of 135 degrees
v has magnitude 400 at an angle of 25 degrees.

3. Vectors and Dot Products
6.4
Vectors and Dot Products
The Definition of the Dot Product of Two Vectors:
The dot product of u = and v = is
The dot product of two vectors is a scalar.

4. Vectors and Dot Products
6.4
Vectors and Dot Products
Examples: Find each dot product.

5. Properties of the Dot Product
Let u, v, and w be vectors in the plane or in space and let c be a scalar.

6. Examples:
Let
Find
First, find u . v
Find u . 2v=
= -28

7. The Angle Between Two Vectors

8. The Angle Between Two Vectors
The angle between two nonzero vectors is the angle , 0    , between their respective standard position vectors, as shown in
Figure This angle can be found using the dot product.
Figure 6.33

9. Example – Finding the Angle Between Two Vectors
Find the angle  between u = 4, 3 and v = 3, 5.
Solution: The two vectors and  are shown in Figure 6.34.
Figure 6.34

10. Solution
This implies that the angle between the two vectors is

11. If is the angle between two nonzero vectors u and v, then
Find the angle between

12. The Angle Between Two Vectors
You try:
Find the angle between

13. The Angle Between Two Vectors
Find

14. The Angle Between Two Vectorsu u u u v v v v u v
Opposite direction
Obtuse
Orthogonal
Acute
Same Direction
Notice that vectors in the same or opposite
direction are multiples of each other.

15. The Angle Between Two Vectors
Figure 6.35 shows the five possible orientations of two vectors.
Figure 6.35

16. The terms orthogonal and perpendicular mean essentially the same thing—meeting at right angles.
Note that the zero vector is orthogonal to every vector u, because 0  u = 0.
Vectors in the same or opposite
direction will be multiples of each other.

17. Orthogonal Vectors (90 degree angles)
The vectors u and v are orthogonal if u . v = 0
Are the vectors orthogonal?
Find the dot product of the two vectors.
Because the dot product is 0, the two vectors are orthogonal.

18. Determine if the vectors u and v are orthogonal, parallel, or neither
Neither, because the dot product is not = 0, and one vector is not a multiple of the other.
Parallel, because u = -4v.

19. Ex:
Use vectors to find the interior angles of the triangle with the given vertices: P (-3,0), Q (2,2), & R (0,6)
Hint: Draw and label the triangle.
Find vectors , , and the angle between them,
Find vectors , , and the angle between them.

20. 6.4 Day 1 Homework:
pg odd

21. 6.4 Vectors and Dot Products Day 2 Vector components and Work

22. Finding Vector Components
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 vector components of u. The vector w1 is the
projection of u onto v and is denoted by w1 = projvu.
The vector w2 is given by w2 = u - w1.
is obtuse
is acute u u w2 w2 v v w1 w1

24. We have seen applications in which two vectors are added to produce a resultant vector. Many applications in physics and engineering pose the reverse problem – decomposing a given vector into the sum of two vector components.
Consider a boat on an inclined ramp

25. The force due to gravity F pulls the boat down the ramp and against the ramp. The 2 orthogonal forces and are the vector components of F.
The negative of represents the force needed to keep the boat from rolling down the ramp and represents the force that the tires must withstand against the ramp.

26. Projection of u onto v
Let u and v be nonzero vectors. The projection of u onto v is

27. w1 = projvu = w2 = u - w1 = u = w1 + w2 =
Ex: Find the projection of onto Then write u as the sum of two orthogonal vectors, one which is projvu. Graph this!
w1 = projvu =
Since u = w1 + w2, then:
w2 = u - w1 =
Check and see:
u = w1 + w2 =

28. You try:
Find the projection of u onto v. Then write u as the the sum of two orthogonal vectors, one which is projvu.

29. You try:
Find the projection of u onto v. Then write u as the the sum of two orthogonal vectors, one which is projvu.

30. Finding Force
A 200 lb. cart sits on a ramp inclined at 30 degrees. What force is required to keep the cart from rolling down the ramp?
To find the force required to keep the cart from rolling down the ramp, project G onto a unit vector v in the direction of the ramp. The projection is

31. Finding Force with a vector projection
A 200 lb. cart sits on a ramp inclined at 30 degrees. What force is required to keep the cart from rolling down the ramp?
To find the force required to keep the cart from rolling down the ramp, project G onto a unit vector v in the direction of the ramp.
The magnitude of this force is lbs of force is required to keep the cart from rolling down the ramp.

32. Finding Force with Right Triangle Trig.
A 200 lb. cart sits on a ramp inclined at 30 degrees. What force is required to keep the cart from rolling down the ramp?
Notice similar triangles (same angles so sides are in proportion) .
We need which is the opposite side from the 30 degree angle in the larger triangle.
The magnitude of this force is lbs of force is required to keep the cart from rolling down the ramp.

33. Finding Force with Right Triangle Trig.
A force of 576 lbs is required to pull a boat and trailer up a ramp inclined at 12 degrees. Find the combined weight of the boat and trailer.

34. Finding Force with Right Triangle Trig.
A force of 576 lbs is required to pull a boat and trailer up a ramp inclined at 12 degrees. Find the combined weight of the boat and trailer.

35. Work
The work W done by a constant force F not directed along the line of motion of an object is given by:
The work W done by a constant force F acting along the line of motion of an object is given by:

36. Ex:
A man pushes a broom with a constant force of 40 lbs. and holds the handle of the broom at an angle of How much work is done pushing the broom 50 ft?

37. You try:
A person pulls a wagon with a constant force of 15 lbs. at an angle of for 500 ft. Find the work done by the person.

38. You try:
To close a barn door, a person pulls a rope with a constant force of 50 lbs. at an angle
Find the work done to move the door 12 ft.

39. Example:
Find the work done in moving a particle from P to Q if the magnitude and direction of the force are given by v.

40. Types of Energy and the units used to measure it
   Many branches of science and technology are involved with energy, and each group originally defined energy using units that they considered useful. It was only much later that these different forms of energy were recognized as different manifestations of the same fundamental quantity. However, the multiplicity of units remains and can be a source of confusion.
One common source of confusion is the difference between mass and weight. Although the two are often used interchangeably, they are fundamentally different quantities. Mass, usually denoted by the letter m, refers to a quantity of matter and is measured in kilograms in the metric system. Weight is the pull of gravity on a mass, and is usually denoted by the symbol w. It is measured in pounds in the British system and Newtons in the Metric system. The two are simply related near the surface of the Earth: to convert mass to weight, you must multiply by the acceleration of gravity, which equals 9.8 m/s2 in the metric system. Thus a mass of m kilograms has a weight of 9.8m Newtons. Note that a body far from the earth may be “weightless,” since the pull of the earth is negligibly small, but its mass has not changed at all from its value at the surface.
Mechanical energy is defined as the work done by a force acting through a distance, and is measured in foot-pounds or Newton-meters, which are called joules. A foot-pound is the work done when a force of 1 pound acts through a distance of 1 foot, and the Newton-meter (or joule) was defined in a similar way. No mechanical work is done if the force does not move the body. Standing still may be tiring, but no physical work is done.

41. Homework: do only on #55 do only 800 meters on #58
pg. 430: , 41, 47, 49, all
do only on #55
do only 800 meters on #58