unit 6

Question 1

cartesian vector vs position vector

Answer 1

• cartesian: endpoints can be defined using cartesian coordinates (x, y)
• position: if cartesian vector was translated so that tail was at origin, then head ould be at some point P(a, b)

Question 2

what’s a unit vector?

Answer 2

length of 1 unit

Question 3

how do u distinguish between a point and a position vector?

Answer 3

position vectors get square brackets

Question 4

what are the two ways to represent position vectors?

Answer 4

OP = [a, b]
OP = ai + bj = a[1, 0] + b[0, 1] (unit vector)

Question 5

relate triangle law of addition to unit vectors

Answer 5

OP = OA + OB = ai + bj

Question 6

what’s i? what’s j?

Answer 6

i = [1, 0] : x-axis
j = [0, 1] : y-axis

Question 7

given [x, y], write using unit vectors i and j

Answer 7

1. break apart into ai + bj
2. turn i and j into square brackets w/ 1 and 0
3. expand a and b into the brackets
4. add both into single square bracket, must be same as given vector

Question 8

given vectors written using unit vectors i and j, write in component form

Answer 8

1. replace i and j w/ square brackets of 1 and 0
2. distribute coefficients
3. add brackets into single final bracket answer

Question 9

whats the formula for finding the magnitude of a cartesian vector

Answer 9

|v| = sqrt((vx2-vx1)^2 + (vy2-vy1)^2)
- finds distance btwn two points

Question 10

rule for adding vectors

Answer 10

v + u = [vx + ux, vy + uy]

Question 11

rule for subtracting vectors

Answer 11

v - u = [vx - ux, vy - uy]

Question 12

rule for multiplying vector by a scalar

Answer 12

if v=[vx, vy]
kv = [kvx, kvy]
*j distribute

Question 13

how do you test if two vectors are collinear?

Answer 13

• if u = kv, plug into (ux/vx) = (uy/vy) nd see if same
• add a scalar multiple to one of them to make them equivalent

Question 14

what does collinear vectors mean?

Answer 14

• vectors can be drawn on same line
• scalar multiples of each other

Question 15

formula for cartesian vector btwn two given points

Answer 15

P1P2 = [x2-x1, y2-y1]

Question 16

find the coordinates and magnitude of each vector. given “AB, for A (x, y) and B (x, y)”

Answer 16

1. label A as x1, y1 and B as x2, y2 order matters
2. plug into AB=[x2-x1, y2-y1]
3. plug vals from 2 into |AB|=sqrt((x^2) + (y^2))
   **acc write formulas

Question 17

geometric vectors in cartesian/algebraic form (have hypotenuse and angle) *force type questions

Answer 17

1. draw diagram
2. write formula F = [|Fx|, |Fy|] or wtv. variable
3. write [Fcosθ, Fsinθ]
4. plug in values nd solve to one decimal place. keep ans. in square brackets

Question 18

what is θ

Answer 18

angle tht vector makes with horizontal OR positive y-axis
- so if u draw something in quadrant 2, the angle is from Q1 to Q2

Question 19

how do u determine resultant velocity?

Answer 19

1. find position vector: r=a+b=[ax+bx, ay+by]=[acosθ+bcosθ, asinθ+bsinθ]
2. find magnitude: |r|= sqrt((x^2) + (y^2)) **plug in whole value
3. find angle: draw |r| magnitude coming out of origin. turn it into a triangle w/ x and y vals from S1. if asking for true bearing, move x up so tht angle is w/ y-axis do tanθ=opp/adj=x/y
4. therefore statement w/ magnitude and angle bearing

Question 20

dot product formula

Answer 20

a*b = |b||proj(b)a|

|proj(b)a| = |a|cosθ ,sooo sub in

a*b = |a||b|cosθ

Question 21

what is the dot product of two vectors? in words

Answer 21

• the product of the magnitudes of vector b and vector a
• vector a is applied in same direction as vector b
• to find magnitude of vector a, we use dot prod’t

Question 22

how must vectors be arranged to use dot product?

Answer 22

tail to tail

Question 23

is dot product a scalar or vector

Answer 23

scalar

Question 24

if θ is greater/less/equal to 90, u*v is…

Answer 24

• θ less than 90: u*v is positive
• θ greater than 90: u*v is negative
• θ equals 90: u*v is zero

Question 25

dot product of two cartesian vectors? given u=[ux, uy] and v=[vx, vy]

Answer 25

u*v = ux*vx + uy*vy - add tgthr to single number after (no brackets) - MUST do dots after plugging in numbers

Question 26

define work + its formula + units

Answer 26

- work is product of magnitude of displacement and magnitude of force applied in direction of motion - |W| = F * (d)cos(θ) - work is in joules

Question 27

formula to find vector projection given magnitude

Answer 27

- geometric formula proj(b)a = |a|cosθ(b̂) - |a|cosθ is scalar, b is vector

Question 28

formula to find vector projection given vectors

Answer 28

- cartesian formula proj(b)a = (a*b / |b|)(b) - |b| can be found by doing b*b

Question 29

formula to find MAGNITUDE of vector projection given vectors

Answer 29

- cartesian formula, but w/o the b-hat nd whole thing is in magnitude proj(b)a = |(a*b / |b|)|

Question 30

formula to find MAGNITUDE of vector projection given magnitudes

Answer 30

if θ is btwn 0 and 90: |proj(b)a|= |a|cosθ(b̂) if θ is btwn 90 and 180: |proj(b)a|= -|a|cosθ(b̂) (in acc, j use 0-90 formula nd say that it is in opposite direction) *b̂ stays as a variable, it represents direction

Question 31

shoe store sold 350 pairs of nike shoes and 275 pairs of adidas. nike sells for $175, adidas sells for $250.

Answer 31

1. cartesian vector to represent shoes sold: s=[350, 275] 2. cartesian vector to represent prices: p=[175, 250] *in order 3. find dot product s*p = revenue

Question 32

the x, y, z axes are ___ to each other

Answer 32

perpendicular

Question 33

how to plot (x, y, z)?

Answer 33

1. start at origin 2. move along each axis, trace path w/ dotted line - x is fwd/bkd - y is left/right - z is up/down 3. draw vector from origin to end of path

Question 34

unit vectors for 3 dimensions

Answer 34

- x-axis: i=[1, 0, 0] - y-axis: j=[0, 1, 0] - z-axis: k=[0, 0, 1]

Question 35

how to find magnitude of vectors in R^3?

Answer 35

|u| = sqrt(a^2 + b^2 + c^2)

Question 36

given A(x, y, z) and B(x, y, z). find magnitude of AB. then, find unit vector, u, in same direction as AB

Answer 36

1. find vector AB using eqn for vector btwn two points. [x2-x1, y2-y1, z2-z1] 2. find magnitude using sqrt formula 1. unit vector is u/|u| 2. equate u/|u| = AB/|AB| 3. multiply reciprocal of magnitude found previously with [vector] found previously. distribute

Question 37

for dot product, when do u use the square brackets?

Answer 37

- only when initially plugging in vectors - no brackets for rest of work AND answer

Question 38

|u-v| where u=[x,y,z] and v=[x,y,z]

Answer 38

1. do vector subtraction to get new coordinate |[x,y,x]| 2. plug into sqrt formula to get magnitude

Question 39

determine if vectors a=[x, y, z] and b=[x, y, z] are collinear using scalar multiples

Answer 39

1. apply a=kb to x, y, and x vals 2. if all 3 isolate for same k-value, it is collinear

Question 40

find a such that [x, y, z] and [x, a, z] are collinear

Answer 40

1. use a=kb for variables u do have to isolate for k 2. multiply k with value u do have to get the value u need

Question 41

formula to find angle btwn two vectors

Answer 41

cosθ = (a*b)/|a||b| - top is dot prod't - bottom is sqrt magnitude

Question 42

find a vector that is orthogonal to [3, 4, 5]

Answer 42

1. uv = 0 2. [x, y, z] * [3, 4, 5] = 0 3. 3x + 4y + 5z = 0 4. find whole # variables - make odd numbers even and isolate for even coefficient

Question 43

how to do right handed rule

Answer 43

- put hand on vector a - see if u can bend fingers twds b - if not, flip hand - thumb up = out of page

Question 44

what is a cross product

Answer 44

value that is perpendicular to vectors a and b such that a x b form a right-handed system

Question 45

betformula for cross product of geometric vectors (given a shape w/ angles nd sides)

Answer 45

axb = (|a||b|sinθ)ň - n with hat: unit vector perpendicular to a and b

Question 46

given two vectors with their magnitudes and angle, find axb and bxa.

Answer 46

1. plug axb into axb = (|a||b|sinθ)ň . the n j. stays. write OUT OF page. 2. write bxa = -axb = -(|a||b|sinθ)ň. plug in vals. youll get same number. get rid of negative and write INTO page.

Question 47

if you've rounded ur final ans, wht must u do?

Answer 47

use squiggly equal sign

Question 48

how to set up cross product given two vector points? formula for cross prod't of algebraic vectors?

Answer 48

a1 a2 a3 a1 a2 b1 b2 b3 b1 b2 axb = [a2b3-a3b2, a3b1-a1b3, a1b2-a2b1]

Question 49

which row goes at the top when doing cross product?

Answer 49

- whichever comes first in question - i.e. bxa means b comes on top

Question 50

a) determine the area of a parallelogram defined by the vectors a=[x, y, z] and b=[x, y, z] b) determine angle btwn the vectors

Answer 50

1. find cross product, ans will be in [brackets] 2. find magnitude, plug into sqrt formula 3. ans in decimal number (no units) 1. axb = (|a||b|sinθ): plug in cross product [x, y, z] into axb 2. do sqrt of |a|and |b| to find magnitude. [x, y, z] = (sqrt of a)(sqrt of b)(sinθ) 3. must find scalar of [x, y, z] vector (bc u cant divide vector/scalar). put into sqrt. formula 4. isolate for theta

Question 51

what is torque?

Answer 51

- measure of the force acting on an object tht causes it to rotate - cross product of force and torque arm

Question 52

torque formulas

Answer 52

T = r x F |T|=|r||F|sinθ - F is force (newtons) - r is the arm (METERS)

Question 53

how do u determine direction of torque vector?

Answer 53

- right hand rule - right hand goes on r vector (arm)

Question 54

given force, angle, and arm length. find magnitude of torque. wht direction does it face?

Answer 54

1. ensure that vectors are tail to tail. redraw if needed, find new angle (z-pattern) 2. convert r to meters 3. plug into formula. ans in unit Nm 4. right-hand rule (hand on r-vector) to determine if into/out of material

Question 55

what kind of quantity is work? its units?

Answer 55

- dot product quantity - joules

Question 56

find work done in direction of travel given force[xyz] and displacement[xyz]

Answer 56

W=F*d

Question 57

how do u find work done JUST against gravity?

Answer 57

- only in z-axis - do dot product BUT with [0, 0, z] and [0, 0, z]

Question 58

triple scalar product

Answer 58

- a*bxc - must perform bxc cross prodt first - then do the dot prodt

Question 59

what is a parallelepiped?

Answer 59

3d figure made from 6 parallelograms

Question 60

how to find volume of parallelepiped

Answer 60

formula: V = |w * u x v| 1. plug in given [bracket vectors] 2. do cross prod't 3. do dot prod't 4. answer in units CUBED (exponent 3) for volume

Question 61

ship/plane Qs tht give two bearings. find algebraic/cartesian vectors. and find resultant velocity.

Answer 61

1. make both diagrams 2. find angle in relation to positive horizontal 3. make vectors as p=[Fcosθ, Fsinθ] and s=[Fcosθ, Fsinθ] 1. r=p + s (resultant = both vectors added tgthr). add them in square brackets 2. find magnitude of square brackets w sqrt formula 3. find angle: use square brackets from S1 to make a right-angle triangle on a graph, then use tan.