Vectors

Question 1

What is a \dot a
What is a x b
When is a \dot b=0, a × b=0
What is |a × b| when a and b are perpendicular
Find cos theta
What is the area of a parallelogram

Answer 1

|a|²
|a||b|sintheta
perpendicular, parallel
|a||b|

norm|a||b|
|a x b|

Question 2

Find the length of the foot of a perpendicular, and length of projection

Answer 2

cross
dot

Question 3

centroid of a triangle

Answer 3

1/3 (a+b+c)

Question 4

Find component of a in n

Answer 4

norm(a dot n)

Question 5

Equation of line in vector, parametric and cartesian form

Answer 5

OA + s AB
(r-a)x(b-a) = 0
x-a1 / b1 = x-a2 / b2 = x-a3 / b3

Question 6

What if b1 = 0 or if b1=b2=0

Answer 6

x = a1

z can be anything

Question 7

How to find intersection between two lines

Answer 7

equate equations

Question 8

Find distance between two lines in one formula

Answer 8

|(b-a)(t x u) | / |t x u|

Question 9

Find foot of a perpendicular
Find length of perpendicular without finding the point (2)

Answer 9

(OF - OC) b = 0
|AC x b| / |b|

Question 10

How to determine the relationship of two lines

Find angle between two lines

Answer 10

parallel, cross, skew

|b1 x b2| / |b1| |b2|

Question 11

Find mirror image of l1 in l2

Answer 11

l2’ : 2(l2 l1) l1 - l2
OA + s AB’

Question 12

Find line where l1 mirrors l1’

Answer 12

s(|b|a - |a|b) + c

Question 13

Find vector, scalar, cartesian, cos plane equations

Answer 13

a + s b + t c
r n = d, d = a n, n = b x c

Question 14

When does a point pass through a plane

Answer 14

dot against n, = d

Question 15

Find perpendicular distance from a point to a plane

Answer 15

(q - a) * n

Question 16

Distance from O to plane

Answer 16

d / n

Question 17

Distance between two planes

Answer 17

d2 - d1 / n

Question 18

Find foot of a perpendicular from a point to the plane

Answer 18

r = OA + s n
(OA + s n) n = d ==> find s

Question 19

How to find relationships of line and a plane

Find angle between a line and a plane

Answer 19

b n = 0
a n = d

theta = cos-1(b n)

Question 20

Find point of intersection between line and plane

Answer 20

(a + sb) * n = d

Question 21

Find angle between two planes

Find relationship between two planes

Answer 21

cos-1(n1 * n2)

intersect / in a line

Question 22

Find the equation for the line of intersection between two planes (both methods)

Answer 22

a + s (n1 x n2)

let x y z = 0 to get a
in a line, sub for s

Question 23

Formulas for Sphere

Answer 23

|r-a| = R
R: radius, a: centre

Question 24

Formulas for Cylinder

Answer 24

|(r-a) - ((r-a)n)n| = R

Question 25

Formulas for Cone

Answer 25

(r-q) n = |r-q|cos a q = centre a = angle from axis

Question 26

Right hand vs left hand

Answer 26

a, b, axb a, b, bxa

Question 27

Scalar triple product and its +/- When is scalar triple product 0

Answer 27

a * (b x c) cyclic -> positive if any 2 are parallel

Question 28

Parallelepiped height and volume

Answer 28

h = a * (b x c / |b x c|) V = a * b x c

Question 29

Vector triple product simplification

Answer 29

ax(bxc) = (a*c)b - (a*b)c

Question 30

For a, b, c that span the space r = 𝛼 a + 𝛽 b + 𝜒 c, what is 𝛼, 𝛽, 𝜒

Answer 30

𝛼 = r * bxc / [a, b, c] 𝛽 - c x a 𝜒 - a x b

Question 31

Define span the space

Answer 31

not zero length / collinear / coplanar

Question 32

Define reciprocal basis What are the common results What is [a, b, c]2

Answer 32

A = b x c / [a, b, c] a A = 1, a B = 0, A r = 𝛼 [bxc, cxa, axb]

Question 33

(reciprocal basis) When are they identical

Answer 33

right handed

Question 34

Test linear independence

Answer 34

p a + q b + r c = 0 ==> see if p=q=r=0 is the only solution

Question 35

Define orthogonal, orthonormal in N dimensions, orthogonal

Answer 35

orthogonal: basis vectors mutually perpendicular orthonormal: basis vectors unit length a*b = summation to n of ai bi

Question 36

Convert to and from cylindrical polar coordinates (list all representations r 𝜃 x y z er e𝜃 ez )

Answer 36

x = r cos𝜃 y = r sin𝜃 z = z er = cos𝜃 i + sin𝜃 j e𝜃 = - sin𝜃 i + cos𝜃 j ez = r

Question 37

Draw a diagram for cylindrical polar

Answer 37

e𝜃 perpendicular er from O

Question 38

What are the ranges of each variable for cylindrical polar

Answer 38

r ≥ 0, 0 ≤ 𝜃 ≤ 2π, -π < 𝜃 ≤ π

Question 39

How does their dot product look like for cylindrical polar

Answer 39

between -1 and 1

Question 40

Convert to plane polar coordinates with a diagram

Answer 40

rcos𝜃 i + rsin𝜃 j = r er

Question 41

Convert to and from spherical coordinates

Answer 41

𝜃 = cos-1 (z / r) φ = tan-1 (y / x) er = sin𝜃 (cosφ i + sinφ j) + cos𝜃 k e𝜃 = cos𝜃(cosφ i + sinφ j) - sin𝜃 k eφ = - sinφ i + cosφ j = er x e𝜃 x = r sin𝜃 cosφ y = r sin𝜃 sinφ

Question 42

Draw a diagram (spherical coordinates)

Answer 42

𝜃 between z and OP φ between x axis to QP

Question 43

Ranges of each variable (spherical coordinates)

Answer 43

r ≥ 0, 0 ≤ 𝜃 ≤ π, -π < φ ≤ π or 0 to 2π

Question 44

Find lengths between two points (spherical coordinates) Find distance from O (spherical coordinates)

Answer 44

PQ = r sin 𝜃 OP = r = r er

Question 45

Find area projection

Answer 45

vector area * n

Question 46

Method to find area projection onto xy plane

Answer 46

A*n = (sx, sy, sz) shoelace - anti consider only plane relevant coordinates