Vectors Flashcards
a x b =
a x b = |a|x|b|sin(θ) n
the vector product
Properties of vector product: a x b = (ka) x b = (a x b) x c = a x (b + c) =
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)
If vectors a and b are parallel:
a x b =
a x a =
0
What are two ways of writing a vector line?
r = a + λd
(r-a) x d = 0
What is the vector area of a triangle?
1/2 |a x b|
How do you convert a vector line to cartesian?
Make (x,y,z) r
Rearrange each coordinate to equal λ
Make equal each other
What are skew lines?
Not parallel lines but they don’t intersect
How do you find the intersection of two vector lines?
Make r coordinates equal each other
Three simulatneous equations with λ and μ
Solve two check with third
Whats the scalar product?
a . b
a . b =
a . b = |a||b| cos θ
How do you find the angle between two vector lines?
a.b=|a||b|cosθ
dot the direction vectors
divide by magnitude of direction vectors
How do you find the shortest distance between a vector line and point?
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
How do you find the shortest distance between two vector lines?
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
What is the equation of a vector plane?
r = a + λd1 + μd2
What is cartesian equation of a plane?
nx + ny + nz = d
Where n is the xyz values of normal vector and d is a.n
How do you convert from cartesian to vector equation of a plane?
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
How do you find the angle between a line and a plane?
θ=90 - φ
cosφ= p.q / |p||q|
With p being direction vector of line and q being normal vector of plane
How do you find the angle between two planes?
cosθ= p.q / |p||q|
With p and q being normal vector of planes
How do you find the distance from a point to a plane?
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
For planes intersecting what are the consistent possibilities?
3 parallel → same plane
2 parallel → solution is a plane
none parallel → sheaf solution is a line
For planes intersecting what are the inconsistent possibilities?
3 parallel → planes stacked above each other
2 parallel → two stacked above one going through both
none parallel → triangular prism
How do you find two planes intersecting?
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
How do you find three planes intersecting?
Find line intersection of two of the planes
Then see where line intersects the third plane
What is the sum of the Eigenvalues?
The sum of the leading diagonal
