Unit 2

Question 1

another way to specify a vector is by

Answer 1

placing it on a Cartesian Plane

Question 2

Cartesian Vector

Answer 2

Vector AB is an example of a “cartesian vector” which has head and tail located on the Cartesian Plane

Question 3

position vector

Answer 3

vector OP = [5, 9] is an example of a “POSITION VECTOR’, which has tail at origin and head at (5, 9)

Question 4

Unit vectors i^ and j^

Answer 4

i^ = [1,0] and j^=[0,1] are examples of “unit vectors” which are vectors that have magnitude equal to 1 unit
i^ and j^ are the “building blocks” for every position vector on the Cartesian Plane
ex: vector OP = [5, 9] (position vector notation) = 5i^+9j^ (unit vector notation)

Question 5

vector OP = [5, 9]

Answer 5

(position vector notation)

Question 6

= 5i^+9j^

Answer 6

(unit vector notation)

Question 7

for any position vector v = [a, b] , a is the magnitude of the ___________ of vector v and b is the magnitude of the ___________of vector v an

Answer 7

horizontal component
vertical component

Question 8

And since vector a= [a, 0] is colinear with i^ and vector b=[0, b] is colinear with ^j, we can write vector v as a linear combination of i^ and j^ in the following way:

Answer 8

vector v = ai^ +bj^

Question 9

since vector a = ai^ = [a, 0] and vector b = bj^ = [0, b] we can also say

Answer 9

vector v = [a, 0] + [0, b]
which is an algebraic way of saying that any vector [a, b] can be written as the sum of its vertical and horizontal vector components

Question 10

direction always measured from

Answer 10

+ x-axis rotating CCW

Question 11

for any cartesan vector can be translated so that its tail is at the ____ and its head is at the point ___

Answer 11

origin (0, 0)
(a, b)

Question 12

in general if A(x, y), and B(x2, y2)
then

Answer 12

vector OP = vector AB = [x2-x1, y2-y1]

Question 13

v^ is a unit vector with the ____ direction as vector v (it has length = __)

Answer 13

SAME
1

Question 14

in generel u^ =

Answer 14

1/|vector u| vector u

Question 15

the dot product of two vectors vector u and vector v is:

Answer 15

vector u (dot) vector v = |vector u||vector v}cos theta where theta is the angle between 0 and 180

Question 16

the result of the dot product is a _____ also called the ___ ___

Answer 16

scalar
scalar product

Question 17

depending on the angle between the vectors, the dot product is either

Answer 17

positive or negative or zero

Question 18

if theta is between 90 degrees + 180 degrees, cos theta < 0, making the dot product

Answer 18

negative

Question 19

Work

Answer 19

the transfer of mechanical energy
the product of the magnitude of the displacement vector and the magnitude of the force vector that is applied in the direction of motion
the units are Newton-meters (N dot m), also known as joules (J)

Question 20

Work =

Answer 20

|vector d| dot (magnitude of the force vector in the direction of the motion)

Question 21

we could have used the dot product to calculate the amount of work done (its the same math) so an alternative definition for work as an application of the dot product is:

Answer 21

work = vector f dot vector d
= |vector f||vector d|cos theta

Question 22

theorum

Answer 22

two non-zero vectors vector u and vector v are perpendicular if and only if vector u dot vector v = 0

Question 23

|vector u||vector v|cos theta = 0 dissect each part

Answer 23

|vector u| (cant be 0) |vector v|(cant be 0) cos theta (cos theta = 0, theta = 90 degrees, vector u + vector v) = 0

Question 24

more properties of the dot product

Answer 24

1. you can multiply by a scalar either before or after taking the dot product, i.e.
   K(vector u dot vector v) = (vector u dot vector v)K for KER
2. you can expand the dot product of a vector with the sum of two other vectors as you could in ordinary multiplication, i.e.
   vector u dot (vector v + vector w) = vector u dot vector v + vector u dot vector w

Question 25

general rule

Answer 25

vector u dot vector u = |vector u|^2

Question 26

if vector u = [a, b] and vector v = [c, d] then:

Answer 26

vector u dot vector v = [a, b] dot [c, d] vector u dot vector v = ac + bd

Question 27

what system do we use

Answer 27

right-handed system grasp the z-axis. your thumb points in the z-direction. your fingers curl from the potitive x-axis toward the positive y-axis

Question 28

3 planes

Answer 28

x-y plane x-z plane y-z plane

Question 29

position vectors in 3-space (or R^3) are represented by

Answer 29

ordered triplets: vector OP = [a, b, c] or using vector notation: vector v = ai+bj+ck where i = [1,0,0], j = [0,1,0], k = [0,0,1], (i, j, k have hats)

Question 30

steps to plot vector v=[a,b,c] in 3-space

Answer 30

1. move a units from the origin in the x-direction 2. then move b units in the y-direction 3. then move c units in the z-direction 4. draw a 3D box showing the 3D nature of the vector 5. draw the vector from the origin to the point Note that each move must be made along a line that is parallel to the corresponding axis

Question 31

addition 3D vectors vector u = [u1, u2, u3] and vector v = [v1, v2, v3]

Answer 31

then vector u + vector v = [u1+v1, u2 + v2, u3 + v3]

Question 32

subtraction 3D vectors vector u = [u1, u2, u3] and vector v = [v1, v2, v3]

Answer 32

then vector u - vector v = [u1-v1, u2 - v2, u3 - v3]

Question 33

to find the vector between two points P1(x1,y1,z1) and P2(x2,y2,z2)

Answer 33

vector P1P2 = [x2-x1, y2-y1, z2-z1]

Question 34

the dot product of two cartesian vectors vector u = [u1,u2,u3] and vector v = [v1,v2,v3] is:

Answer 34

vector u dot vector v = u1v1 + u2v2 + u3v3

Question 35

given 2 noncolinear 3D vectors the cross product produces a

Answer 35

3rd vector that is perpendicular to both of the first two vectors

Question 36

any two vectors alwayse

Answer 36

determine some plane (2D surface)

Question 37

how many vectors can be perpendicular to a plane

Answer 37

indefenently many

Question 38

the cross product is how you find

Answer 38

a specific vector that is perpendicular to two noncolinear vectors (relative to their tails)

Question 39

cross vector formula

Answer 39

vector a cross vector b = [a2b3-a3b2, a3b1-a1b3, a1b2-a2b1]

Question 40

k(vector a cross vector b) is ___

Answer 40

perpendicular to vector a & vector b

Question 41

dot product to verify the perpendicularity

Answer 41

vector a cross vector b dot vector a vector a cross vector b dot vector b

Question 42

vector b cross vector a =

Answer 42

-(vector a cross vector b)

Question 43

direction of the cross product vector

Answer 43

our cross product gets a particular vector in a pre-determined direction. the right hand rule: direction, point 1st vector, curl 2nd vector, thumb direction of the cross product vector

Question 44

magnitude of the cross product of 2 vectors vector a and vector b is:

Answer 44

|vector a cross vector b| = |vector a||vector b|sin theta

Question 45

cross product of two vectors in 3D is a vector with magnitude

Answer 45

and is perpendicular to the two vectors such that they form a right handed system

Question 46

cross product of 3 vectors

Answer 46

vector a cross vector b cross vector c will also produce a vector

Question 47

we can also combine the cross product with the dot product:

Answer 47

"triple scalar product" vector a cross vector b dot vector c will produce a scalar

Question 48

could you do the cross product of a vector with itself

Answer 48

no, a vector would be collinear with itself

Question 49

does the cross product of 2-D vectors make sense

Answer 49

no, you would need a 3rd dimension