unit 7

Question 1

what do vector equations vs standard form/y-int form eqns define?

Answer 1

• y-int/standard form: define all points (x, y) on line
• vector: eqn tht describes resultant vectors (start at origin, end at point on line)

Question 2

vector eqn of a line

Answer 2

[x, y] = [x0, y0] + t [m1, m2]
- [x0, y0] is any pt on line
- [m1, m2] is direction vector parallel to line

Question 3

for a line tht goes through the point A(x, y) and B(x, y), write a vector eqn.

Answer 3

1. find slope with AB = [x2-x1, y2-y1] and sub into [m1, m2]
2. plug either given point into [x0, y0]

Question 4

determine 3 more position vectors given a vector equation

Answer 4

1. plug in literally any number for the t-values and expand into [m1, m2]
2. actually ADD [x0, y0] + [m1, m2]

Question 5

determine if the point (x, y) is on the line of a vector equation

Answer 5

1. write down given vector eqn
2. plug pt into [x, y]
3. split into two parametric eqns (one for x, one for y) and isolate for t in both
4. if t is the same, pt is on line. if not, pt not on line.

Question 6

given two parametric eqns, find the coordinates of a point on the line

Answer 6

1. plug in literally any value for t in both eqns (same t val in both)
2. put into round brackets (x, y)

Question 7

write a vector eqn of a line given the parametrics eqns

Answer 7

1. find a coordinate and plug into [x0, y0]
2. get ur [m1, m2] from #t in parametric eqns
3. simplify slope if possible

Question 8

scalar eqn

Answer 8

Ax + By + x = 0

Question 9

write scalar given parametric eqns

Answer 9

1. isolate t in both eqns
2. set t equal to each other
3. rearrange to get to standard form (trick is to do y=, THEN move y to other side)

Question 10

determine if two parametric eqns are parallel (l1 and l2?)

Answer 10

1. state slopes of each, like ↑m1=[m1, m2] and ↑m2 = [m1, m2]. *reduce both
2. write ↑m1=k↑m2 an dplug in [m1, m2] for both.
3. split into x and y. isolate for k in both.
4. if k (scalar multiple) is same, parallel.

Question 11

graph a scalar equation

Answer 11

1. find x-intercepts by setting y to 0
2. find y-intercept by setting x to 0
3. plot both points and connect w/ ruler
4. label w/ eqn

Question 12

find position vector perpedicular to line from scalar equation

Answer 12

METHOD ONE
1. use intercepts to find slope. do ↑AB=[Bx-Ax, By-Ay]
2. find negative reciprocal of slope
3. plot pt, connect from origin
METHOD TWO
1. Ax + By + C = 0. perpendicular vector is [A, B]
2. plot nd connect from origin

Question 13

make a scalar eqn given vector eqn

Answer 13

METHOD ONE
1. split into x nd y eqns. then isolate for t
2. set t=t and set to 0.
METHOD TWO
1. reciprocate [m1, m2] and write ↑n=[m1, m1]. (dot prod’t=0) **n has right angle symbol in subscript to rep. perpendicular
2. sub in [x0, y0] for x and y in m1x+m2y+c=0 to isolate for c
3. write final eqn

Question 14

how can lines in three-space be defined?

Answer 14

• vector eqn
• parametric eqn
• symmetric eqn
• NOT scalar eqn bc scalar eqns define planes

Question 15

wht is a plane

Answer 15

2d flat surface tht extends infinitely in all directions

Question 16

how do vector nd parametric eqns change in R^3 (three-space)?

Answer 16

add additional for z

Question 17

symmetric eqn

Answer 17

(x-x0)/m1 = (y-y0)/m2 = (z-z0)/m3
- basically, isolate for t in all parametric eqns, then set equal t=t=t

Question 18

what 4 scenarios do u need to write the eqn of a plane?

Answer 18

things needed: two non-collinear vectors parallel to plane + a point on plane
1. line and a point not on line
2. three non-collinear pts
3. two intersecting lines
4. two parallel, non-coincident lines

Question 19

how can u write eqn of a plane?

Answer 19

• vector
• parametric
• scalar

Question 20

vector eqn of a plane?

Answer 20

↑r = ↑r0 + t↑a + s↑b

[x, y, z] = [x0, y0, z0] + t[a1, a2, a3] + s[b1, b2, b3]

• ↑a and ↑b are non-parallel direction vectors parallel to plane
• t and s are scalars

Question 21

write vector and parametric eqns of a plane given two direction vectors and a point

Answer 21

1. write ↑r = ↑r0 + t↑a + s↑b
2. [x0, y0, z0] is point given
3. t[a1, a2, a3] + s[b1, b2, b3] are the direction vectors given. j write in values. leave t and s
4. split into parametric. MUST write pi{x=, y=, z=

Question 22

determine if the pt (x, y, z) is on a plane given parametric eqns

Answer 22

1. use elimination/substitution to solve for t with x/y parametrics
2. plug t into x-parametric eqn to solve for s
3. plug s and t into x and y parametrics as LS=RS check
4. do LS/RS check with z-parametric eqn
5. since t=# and s=# satisfied/didnt satisfy allll eqns, pt Q is/isn’t on plane

Question 23

find coordinates of two other points on plane given vector eqn

Answer 23

1. plug in any value for t and s (state them beforehand)
2. expand until u get one final [x, y, z]
3. put into round brackets

Question 24

find x-intercepts of a plane given vector eqn/parametric eqn

Answer 24

1. write out parametric eqns
2. sub in (x, 0, 0) into all parametrics
3. isolate either t or s (wtvr gives whole numer) in y or z parametric. do substitution/elimination to find t and s values.
4. take the s and t vals, plug them into x parametric eqn to get x-intercept
5. answer as “x-intercept is (x-val, 0, 0)”

Question 25

find vector and parametric eqns of each plane given x-int, y-int, and z-int

Answer 25

1. write intercepts as points. A(x, 0, 0), B(0, y, 0), C(0, 0, z) 2. write vectors of ↑a=↑AB and ↑b=↑AC. then find the vals by subtracting each coordinate (like, Bx-Ax) 3. put ↑a and ↑b into vector equation. use any intercept point as [x0, y0, z0] 4. make parametrics as pi{

Question 26

given two vectors eqns as such: [x, y, z] = [x0, y0, z0] + t[a1, a2, a3] [x, y, z] = [x0, y0, z0] + s[b1, b2, b3] make a vector eqn

Answer 26

1. check tht lines arent parallel by finding scalar multiples. [a1, a2, a3] = k [b1, b2, b3] 2. plug either pt for [x0, y0, z0]. then use t[] and s[] from BOTH eqns

Question 27

scalar eqn of a plane

Answer 27

Ax + By + Cz + D = 0 where ↑n = [A, B, C] is normal vector to the plane

Question 28

is vector [x, y, z] parallel to the plane? given normal vector and point of plane.

Answer 28

1. do dot product of vector ur testing and the normal vector 2. if 0, testing and normal are perpendicular, meaning tht testing vector is parallel to plane

Question 29

is vector [x, y, z] normal to the plane? given normal vector and point of plane.

Answer 29

1. set vectors equal to each other as [b]=k[n] 2. isolate for k 3. if k is same, then scalar multiple exists nd vector b is normal to the plane bc its collinear w/ the normal

Question 30

find scalar equation of a plane given 3 points

Answer 30

1. find vectors AB and AC (w/ the subtracting stuff) 2. cross product of AB and AC. reduce if possible 3. plug vector into ABC in eqn, xyz is any point. isolate for D 4. write final eqn

Question 31

determine scalar eqn of plane parallel to xz-plane, through a certain point.

Answer 31

1. in xz plane, so normal is [0, 1, 0] 2. plug normal vector and given point into Ax+By+Cz+D=0 and isolate for D 3. make final eqn (some terms may fullt cancel)

Question 32

determine scalar eqn of plane containing a line with t[ ] and perpendicular to s[ ]

Answer 32

1. use point from line contained as point on plane (P0) 2. use s[ ] from perpendicular as normal 3. plug into Ax+By+Cz+D=0, isolate for D 4. make final eqn

Question 33

possibilities of intersection in 2-space

Answer 33

- intersection at 1 point: eqns have diff slopes/negative reciprocals - infinite poi: same slope, same b - no poi: same slope, diff b

Question 34

find the solutions of each system. given l1 and l2 in standard form. Ax-By=C

Answer 34

1. use elimination, make eqns similar. 2. if both eqns cancel fully, infinite poi. if 0=not zero nmuber, then parallel lines (no poi)

Question 35

find the solutions of each system. given l1 and l2 in vector eqn form. [x, y] = [x0, y0] + s/t [a1, a2]

Answer 35

1. determine type of line (parallel/not parallel)by looking at both eqns first. (if u can reduce slopes into each other, its parallel). **scalar multiples 2. make parametric eqns. 3.equate x=x an y=y 4. rearrange, set to 0 5. solve for s and t using sustitution or elimination. if 0=0, it's coincident. if 0=#, it's distinct. IFFF u had a not parallel line, u will isolate for t. plug t into equated parametric eqns and solve for s. 6. CHECK: plug s and t into original parametric eqns. expand, put into round brackets. both should give same coordinate. this is the poi.

Question 36

possibilities of intersection of lines in 3-space

Answer 36

- no intersections (parallel/distinct) - no intersections (not parallel, skewed) - 1 intersection (not parallel, on same plane) - infinite poi (parallel, coincident)

Question 37

how to solve intersection of lines in 3-space

Answer 37

1. do scalar vectors. check if parallel or not. (if all k-vals same, parallel) PARALLEL: check if a given point is on both lines. split l2 into parametric eqns, isolate for t. if t-vals are same, it is coincident. if t-vals are diff, its distinct) NOT PARALLEL: solve system w/ elimination/substitution after doing x=x and y=y w/ parametrics (compare y and z eqns). this will give u t and s vals. plug s and t vals into x=x parametric. if x=x, then there is 1 POI. if no POI, it is skewed.

Question 38

what is the shortest distance btwn two skew lines?

Answer 38

- length of common perpendicular d= (↑P1P2↑n) / |↑n| P1 nd P2 r any points on each line ↑n = m1 x m2 is the normal for both lines

Question 39

determine the distance btwn the skew lines

Answer 39

1. find ↑P1P2 = [ ] w/ subtraction stuff. P2 - P1 2. do cross product with m1 nd m2 from original l1 and l2 to find n 3. find |↑n| w/ square root stuff 4. put all into equation. d= (↑P1P2↑n) / |↑n| 5. answer in acutal number and units

Question 40

possibilities of intersections of lines and planes

Answer 40

- intersect at one point (line goes thru plane) - infinite solutions (line lies on plane) - no solutions (line is parallel nd distinct from plane

Question 41

determine if line and plane intersect. if so, find solution. given pi1 eqn in standard form for plane and parametric eqn for line.

Answer 41

1. sub in entire x, y, z parametric formulas into x, y, z in standard form eqn. expand and isolate for t. t=# means one solution 0=# means no solution 0=0 means infinite solutions 2. plug t into parametric eqns to find solution. put into round bracket as (x, y, z)

Question 42

determine if a line intersects the plane without solving. given pi1 standard form eqn and vector eqn of line.

Answer 42

1. find ↑n using coefficients from standard form eqn = [xyz] 2. get ↑m from s[ ] of l1 eqn 3. do dot product. - if it equals 0, normal and vector r perpendicular. thus, line doesnt intersect plane - it it doesnt equal 0, normal and vector not perpendicular, so lin intersects plane

Question 43

find the distance between plane and point. given plane eqn in standard form and a point.

Answer 43

1. find point Q. j. pick wtvr two numbers for A and B, then isolate for z. you will have final point. (try zeroes) 2. find vector ↑PQ w/ subtracty stuff. 3. ↑n is coefficients of standard eqn of plane. 4. plug into d= (↑PQ*↑n) / |↑n|. solve, add units.

Question 44

possibilities for intersection of planes

Answer 44

- line of intersection (set of points that lie on line of intersection) - infinite solutions (planes are coincident, all points on plane r solution) - no solutions (parallel and distinct planes dont intersect)

Question 45

how to figure out how planes intersect given two scalar/standard plane eqns.

Answer 45

1. write both as normal vectors using ↑n=[ABC] 2. check scalar multiples. PARALLEL: do elimination - if 0 = #, no solutions - if 0=0, infinite solutions NOT PARALLEL: - intersect in a line