Vectors

Question 1

a x b =

Answer 1

a x b = |a|x|b|sin(θ) n

the vector product

Question 2

Properties of vector product:
a x b =
(ka) x b = 
(a x b) x c =
a x (b + c) =

Answer 2

a x b = -b x a
(ka) x b = k(a x b)
(a x b) x c = a x (b x c)
a x (b + c) = (a x b) + (a x c)

Question 3

If vectors a and b are parallel:
a x b =
a x a =

Answer 3

0

Question 4

What are two ways of writing a vector line?

Answer 4

r = a + λd

(r-a) x d = 0

Question 5

What is the vector area of a triangle?

Answer 5

1/2 |a x b|

Question 6

How do you convert a vector line to cartesian?

Answer 6

Make (x,y,z) r
Rearrange each coordinate to equal λ
Make equal each other

Question 7

What are skew lines?

Answer 7

Not parallel lines but they don’t intersect

Question 8

How do you find the intersection of two vector lines?

Answer 8

Make r coordinates equal each other
Three simulatneous equations with λ and μ
Solve two check with third

Question 9

Whats the scalar product?

Answer 9

a . b

Question 10

a . b =

Answer 10

a . b = |a||b| cos θ

Question 11

How do you find the angle between two vector lines?

Answer 11

a.b=|a||b|cosθ
dot the direction vectors
divide by magnitude of direction vectors

Question 12

How do you find the shortest distance between a vector line and point?

Answer 12

Write p in terms of λ
Find vector from point to ‘p’
If you dot them with direction vector it will equal 0 as perpindicular so solve for λ
Then find magnitude of vector

Question 13

How do you find the shortest distance between two vector lines?

Answer 13

Write ‘p’ in terms of λ and ‘q’ in terms of μ
Find vector from p to q
Dot them with two dircetion vectors for simultaneous equations for λ and μ
Substitude and find magnitude of vector p to q

Question 14

What is the equation of a vector plane?

Answer 14

r = a + λd1 + μd2

Question 15

What is cartesian equation of a plane?

Answer 15

nx + ny + nz = d

Where n is the xyz values of normal vector and d is a.n

Question 16

How do you convert from cartesian to vector equation of a plane?

Answer 16

dot n with vector ‘p’
assume two values and substitute p in for third then solve for p this is one direction vector
dot n with vector ‘q’
assume two different values and substitute q in for third then solve for q this is the second

Question 17

How do you find the angle between a line and a plane?

Answer 17

θ=90 - φ
cosφ= p.q / |p||q|
With p being direction vector of line and q being normal vector of plane

Question 18

How do you find the angle between two planes?

Answer 18

cosθ= p.q / |p||q|

With p and q being normal vector of planes

Question 19

How do you find the distance from a point to a plane?

Answer 19

Find equation through point perpindicular to plane using n = d
Find point of intersection using coordinates of r and substituting into cartesian plane equation to find λ
Then find p to point vector and magnitude

Question 20

For planes intersecting what are the consistent possibilities?

Answer 20

3 parallel → same plane
2 parallel → solution is a plane
none parallel → sheaf solution is a line

Question 21

For planes intersecting what are the inconsistent possibilities?

Answer 21

3 parallel → planes stacked above each other
2 parallel → two stacked above one going through both
none parallel → triangular prism

Question 22

How do you find two planes intersecting?

Answer 22

Cross two normal vectors to get direction vector

Then need point on line so choose x value and solve for y and z with cartesian plane equations

Question 23

How do you find three planes intersecting?

Answer 23

Find line intersection of two of the planes

Then see where line intersects the third plane

Question 24

What is the sum of the Eigenvalues?

Answer 24

The sum of the leading diagonal