U7L1:Sec. 7.1/7.2 Vectors As Forces & Velocity
MCV4ULessons
U7L1:Sec. 7.1/7.2 Vectors as Forces & Velocity
When we think of force, we think of things like pulling an object, lifting something, throwing. However, these are all examples of effort or muscular exertion. What about the attraction of the moon to the Earth or the attraction of metal to a magnet.
Force can be defined as that which changes, or tends to change, the state of rest, or uniform motion of a body. Force is a ______and therefore must be described using both ______. On Earth, force is defined as the product between the mass of an object and the acceleration due to gravity (9.8 m/s2). So a 1 Kg mass exerts a downward force of 1 Kg x 9.8 m/s2 or 9.8 . The unit of measure for force is called ______and is abbreviated as N. Therefore a 1 Kg mass exerts a force of 9.8 N downward.
Often when looking at a problem involving forces, there are many forces acting on an object. It is useful to be able to look at a single force called ______which is the sum of all the forces working on the object. The process of finding the resultant of all the forces acting on an object is called the ______. If a single force that when applied to the object prevents any further motion than the object is said to be in ______. The net force on the object is the zero vector.
Eg.1: Two boys push a box to the east with forces of 30 N and 40 N respectively. What is the net force.
At third boy decides to try and stop the box from moving and exerts a force of 55 N to the west. What is the net force.
Eg.2: Two forces act on a large object. 500 N are used to pull the object south. 800 N are used to push the object on a bearing of 100º. What is the resultant force?
Resolving a Vector into Components
In many situations involving forces, we are interested in a process that is the______. This process is called______, which means taking a single force and decomposing it into two components.
Resolve vector into its horizontal and vertical components.
Eg.3: A car is traveling at 80 km/h on a bearing of 260º, draw a vector to represent the car’s velocity and write the velocity in component form.
Eg.4: A mass of 20 Kg is suspended from a ceiling by two ropes that make angles of 60º and 45º with the ceiling. Determine the tension in each of the ropes.
Homework: P.363 # 5a , 8 , 12 , 13 , 15 , 17P.369 # 2 , 4 , 5 , 8
U7L2: Sec. 7.3 The Dot Product of Two Geometric Vectors
The dot product is the ______of 2 vectors. The result is a______.
The dot product for any two vectors is defined as the product of their magnitudes multiplied by the cosine of the angle between the two vectors when the two vectors are placed in a tail-to-tail position.
Eg.1: If and , find if the angle between them is 750 , 900 and 1300.
When is acute, ______
When is 900, ______therefore are______.
When is obtuse, ______
Properties of the Dot Product
- Commutative Law:
We can change the order in the multiplication because the quantities are just scalars in the formula.
- Distributive Law:
- Other Properties:and
- Associative Property with a scalar k:
Eg.2: If the vectors are perpendicular, and , determine the angle (to the nearest tenth of a degree) between the nonzero vectors .
Homework: P.377 # 6acf , 7bde , 8 , 9 , 11 - 15
U7L3: Sec. 7.4 The Dot Product of Algebraic Vectors
In R2 , if then______.
In R3 , if then______. Proof in textbook p.379
Eg.1: If then;
a) Determine
b) Find the angle between .
Eg.2: Determine the value of “k” such that are perpendicular given .
Eg.3: Evaluate the following;
a)b)c)d)e)f)
Eg.4:Find a vector (or vectors) perpendicular to each of the vectors
Homework: P.385 # 2 , 3 , 6bc , 7a , 9b , 10a , 11 - 14 , 17**
U7L4: Sec. 7.5 Scalar and Vector Projections
When two vectors, , are placed tail to tail, and is the angle between the vectors, , the scalar projection of is ON. You can think of a projection like a shadow with the light shinning down from above. Ie: the shadow that casts on .
If the angle is______, then the scalar projection is ______and if the angle is ______than the scalar projection is ______. The scalar projection of vector ______is not equal to the scalar projection of______. Another observation is that the scalar projection of vector is independent of the length of .
Calculating the scalar projection of :
We need to solve for using the dot product formula.
Since then
. Similarly, .
Eg.1: Determine the scalar projection of vector if .
The calculation of the vector projection of is the corresponding scalar projection of multiplied by which is a unit vector pointing in the direction of .
Eg.2: Determine the vector projection of vector if .
Directional Cosines:
Eg.3: The point R ( -5 , 3 , -3 ) is the corner of a box with another corner sitting at the origin in R3.
a) Determine the .
b) Determine the directional cosines for .
Homework: p. 398 # 1 , 6 , 7c , 11 , 13a , 15ab , 16** , 17**
U7L5: Sec. 7.6 The Cross Product of Two Vectors
When we determined the dot product of two vectors the result was a______. The cross product of two vectors is a ______.
The Magnitude of the Cross Product:
To find the direction of the cross product we use the Right-Hand Rule.
- Direct the fingers of your right hand in the direction of .
- Curl your fingers towards .
- Your thumb will point in the direction of .
- The result of is a new vector that is perpendicular to the plane in which the given vectors lie.
Eg.1: Determine then for the following situations;
a)b)
n.b. but the directions will be opposite
The Cross Product for Cartesian Vectors
For then ______
Trick: Write the components of the vectors in 2 rows starting with the 2nd component and repeating it at the end. Take the downward product minus the upward product.
Eg.2: Find given .
For collinear vectors,
Combining the dot and cross products: implies because would not be defined since the result would be a scalar crossed with a vector. We can use this property to test if 3 vectors are coplanar.
If are coplanar, then is perpendicular to both and since are in the same plane, ______.
Eg.3: Determine if the following vectors are coplanar.
Homework: p.408 # 4bcef , 5 , 8 , 9 , 12 , 13
U7L6: Sec. 7.7 Applications of the Dot Product and Cross Product
When a force is acting on an object so that the object is moved from one point to another, we say that the force has ______.
Eg.1: A rickshaw driver pulls his cart at an angle of 30º with the horizontal. If his current customer requires
him to pull up on the rickshaw handles with a force of 60 N, and the trip is a total of 750 m, how much work, in J, has the rickshaw driver done over the course of this trip?
The cross product of two vectors, defined by can be used to find the area of a parallelogram.
N.B. A triangle is half of a parallelogram along the diagonals.
Eg.2: Determine the area of with vertices Q ( 2 , 4 , 1 ) , R ( 3 , -3 , 5 ) and S ( -2 , 0 , 2 ).
The cross product can also be used in the consideration of forces that involve rotation, or turning about a point or an axis. The rotational or turning effect of a force is called ______. is the force applied to the lever (or wrench). is the distance from the pivot point to the point where the force has been applied. Its tail is at the pivot point and its head at the point where is applied. is the angle between .
The unit for torque is the______.
Eg.3: A force of 30 N is used to open a hinged lid on the top of a box. The force is applied 30 cm from the hinge at an angle of 45º. Calculate the magnitude of the torque.
Homework: P.415 # 3 , 5b , 6 , 7a , 8 , 9
U7L7: Unit # 7 – Review P.418 # 2 – 7 , 8b , 9 - 12 , 15ab , 16 , 18 , 20 , 22 - 25 , 27ab , 30 , 31ab , 33 , 34
P.422 #1c 2abc 3 4 5ab 6 7
U7L8: Unit # 7 Test
Page 1 of 12Prepared by: Scott McEwen