6.1 Vectors in the Plane PreCalculus
6 - 1 6.1 VECTORS IN THE PLANE
Learning Targets:
1. Find the component form and the magnitude of a vector. 2. Perform addition and scalar multiplication of two vectors. 3. Sketch vectors and resultant vectors.
4. Use vectors to solve problems.
5. Write a vector in terms of the standard unit vectors i and j.
Definition: A vector is a directed line segment that has _____________________ and ____________________. Each vector has an initial point and a terminal point.
Two vectors are ____________________if they have the same magnitude and direction.
The location of the vector does not matter. A vector in standard position has its initial point at _______________. If vector v is in standard form and has a terminal point (v1, v2), then the component form can be written as ___________. The magnitude of a vector v is denoted __________. You can use the distance formula to find the magnitude.
When the vector is in component form a b, , the magnitude is simply ______________________________. Example 1: Let P (4, 1) and Q (10, 6).
a) Sketch vector PQ and find its magnitude.
b) Draw the vector in standard position and write its component form.
c) Show algebraically why vector PQ is equivalent to the standard position vector.
A more precise way to specify the direction of a vector is to give the direction as an angle measurement.
Example 2: Consider the vector below with magnitude r and direction θ. If the endpoint of the vector is (x, y) what is the component form of the vector in terms of r and θ ?
θ r
6.1 Vectors in the Plane PreCalculus
6 - 2
Example 3: Sketch vector v in standard position with magnitude 41 and direction 220.
Example 4: Find the component form of the vector in the previous example.
Vector Operations
The two basic vector operations are vector addition and scalar multiplication. Scalar multiplication is multiplying a vector by a real number (aka, a “scalar”). When two vectors are added together the “answer” is called the resultant vector. In the paper toss game, we wanted the resultant vector to have a magnitude and a direction that would land the paper in the trash can. We had to add the velocity vector for the paper toss to the velocity vector of the wind for that to happen.
Example 5: Given u = 2, 3- and v = - -5, 2 . Find the component form of each resultant vector.
a) 2u + v b) u – v
Example 6: Find the component form of the resultant vectors from the previous examples algebraically.
Example 7: Forces with magnitudes of 2000 Newton and 900 Newton act on a machine part at angles of 30 and –45 respectively. Find the magnitude of the resultant vector of these two forces.
6.1 Vectors in the Plane PreCalculus
6 - 3 Unit Vectors
A unit vector is a vector in the same direction but with magnitude = _______. How do you make something with a length of 10 become a length of 1?
If the magnitude of a vector is m and the component form of the vector is a b, , then the unit vector is _______________.
: Officially, the unit vector u of a non-zero vector v is found using u= v
v … in case you were wondering . Example 8: Find the unit vector of 5,1 .
Standard Unit Vectors
The two unit vectors i = 1, 0 and j = 0,1 are called standard unit vectors. Any vector can be written as linear combination of these unit vectors. In other words, if v = c d, , then v = ci + dj.
Example 9: Let u be the vector with initial point (2, –5) and terminal point (–1, 3). a) Find the component form of vector u.
b) Write u as a linear combination of the unit vectors i and j.
Example 10: Find the unit vector for –i – 3j.
To summarize vector notation…Vectors can be written in the following ways: 1. When typed, we use a bold font … u
2. When written by hand, we use a “half arrow” above the vector … u or AB 3. We can use component form (which implies the vector began at the origin) … c d, 4. We can use standard unit vectors … ci + dj
6.2 Dot Product of Vectors Pre-Calculus
6 – 4 6.2 DOT PRODUCT OF VECTORS
Learning Targets:
1. Find the dot product between two vectors. 2. Use the dot product to find the length of a vector. 3. Find the angle between two vectors.
4. Show vectors are orthogonal.
5. Determine whether vectors are parallel, perpendicular, or neither. 6. Find work done by a force moving on an object.
Dot Product: The dot product of u = u u1, 2 and v = v v1, 2 is u v =u v1 1+u v2 2. • Notice the dot product of two vectors produces a ____________________. • The dot product is also called the inner product.
• Also gives us another way to find the length, or magnitude, of a vector because of the property u u =
u
2. • Orthogonal Vectors are two vectors whose product is ___________.Example 1: Given vectors: a = 2, 5− , b = 4,1 and c = 10, 4 . a) Find a b and a c .
b) Which two vectors are orthogonal?
Sketch the two orthogonal vectors…what do you notice?
Example 2: Use the dot product to find the length of the vector u = −8, 15 .
Angle between two vectors: The angle, θ, between any two non-zero vectors is the corresponding angle between their respective vectors in standard position. The cosine of this angle, θ, equals the dot product of the two vectors divided by the product of the magnitudes of these two vectors.
6.2 Dot Product of Vectors Pre-Calculus
6 – 5 Example 3: Sketch each vector. Find the angle between the vectors.
a) u = 〈–5,3〉, v = 〈–4,7〉 b) a=2i−5j , c=10i+4j
Besides helping us find the angle between two vectors, the dot product is useful in physics to calculate the amount of work needed to move an object from an initial point to a final (terminal) point.
Work: In general, work is defined as force multiplied by a distance.
• When F is a force whose direction is the __________ as the distance an object is moving, we find work by…
• When F is a force in __________ direction, we find work by…
Example 4: Find the work done when lifting a 87-pound weight 35 feet into the air.
Example 5: Suppose AB = 2i + 3j while the angle between AB and a 170-pound force F is 30°. Sketch a picture for this
situation and find the work done by F in moving the object from A to B.
6.3 Parametric Equations and Motion PreCalculus
6 - 6 6.3 PARAMETRIC EQUATIONS AND MOTION Learning Targets:
1. Graph parametric equations by hand and with calculator. 2. Eliminate the parameter to write Cartesian function. 3. Write parametric equations.
4. Solve problems involving parametric equations. Day 1
Try this Exploration:
1. Press “mode” and make sure that the calculator is set to Degree and Par (for parametric mode)
2. Press “y=“ and enter cos(T) for X1T and sin(T) for Y1T. To get the
variable “T” press the variable “x,T,Θ,n” button.
3. Press “window” and set
Tmin = 0 Tmax = 360 Tstep = 2 Xmin = –2 Xmax = 2 Xscl = 1 Ymin = –2 Ymax = 2 Yscl = 1
4. Press “trace”. Use the arrow keys to move the cursor left and right. What did we find? What functions made this picture? How do these functions relate to the Pythagorean identity? Can we write this function any other way?
Definition: Parametric Curve
A parametric curve is the graph of ordered pairs (x, y) where
( )
x
=
f t
,y
=
f t
( )
are functions defined on an interval of t values. Parametric equations are a set of equations where the horizontal component and the vertical component are defined according the same variable or parameter.
6.3 Parametric Equations and Motion PreCalculus
6 - 7
Example 1: Create a table of values and the graph the pair of parametric equations:
x
= +
t
4,
y
=
t
2Example 2: Eliminate the parameter to write a Cartesian equation to identify the graph of the curve.
a) x = t + 4, y = t² b) x = 3cos t, y = 3sin t
Day 2
On the homework for 6.3 day 1, we found parametric equations for a line (or line segment) when given two ordered pairs. When given a Cartesian equation, the easiest way to create parametric equations is to simply let x = t and substitute t in for every x so that y is a function of t.
Example 3: Find a parameterization for the curve y= −16x2
+
21
x+6–10 10
10
x y
6.3 Parametric Equations and Motion PreCalculus
6 - 8
Parametric Equations and Projectile Motion
Earlier this year, we used 2
0 0
( ) 16
h t = − t +v t+ to model the height of an object at a given time. The graph h generated by this equation is no different for an object thrown straight up than for an object thrown across the room. We need to redefine 2 0 0 16 x t y t v t h = = − + +
using a velocity vector in component form _____________________A more accurate projectile motion equation is…
Example 4: A baseball is hit 129 feet per second at angle of 30° relative to level ground from an initial height of 3 feet.
a. Write the two parametric equations that model the path of the ball as a function of time.
b. After 0.5 seconds, how far has the ball traveled horizontally and vertically?
