Vectors Flashcards
3D vector equation
( a ) ( a1 )
r = ( b ) + λ ( b1 )
( c ) ( c1 )
r = position vector of a point of a line + λ(direction)
Simplifying a 3D vector
( a + λa1 )
r = ( b + λb1 )
( c + λc1 )
Vector equation of a line that passes through A and B
—->
AB = B - A
—->
r = A + λ(AB)
Show a vector can be rewritten in a different way
Show one of the directions is another multiplied by a scalar (parallel)
Let λ = 0 and find the μ value that gives the same point to find a common point
Finding the points on a line a distance from a point
Simplify the vector in terms of a + λa1…
Use pythagoras on the x,y and z and square
Make a quadratic in terms of λ
Solve for λ and substitute into the original vector equation
Vector in Cartesian form
Rearrange each In terms of λ
x - a1 y-a2. z-a3
——— = ——— = ———
b1. b2. c2
Cartesian to vector
Set each part of the Cartesian form equal to λ and rearranged for x,y, z
Write back into Cartesian form
Equation of a plane
r = a + λb + μc
Where A is a point on the plane and b is the direction vector from A to B and c the direction vector from A to C
Cartesian equation of a plane
n1x + n2y + n3z = p
Scalar/dot product
a.b
Multiply the x coefficients together, repeat with y and z then sum
Angle between two vectors
a.b
cos θ = ———–
|a||b|
Take the mod of the whole thing if you want an acute angle
How to see if two vectors are perpendicular
a.b = 0
Finding a possible vector
Set z = 1, solve for x and y then scale
Angle between two lines
a.b
cos θ = ———–
|a||b|
Where a and b are the direction vectors of two lines
Equation of a plane
r.n = p
Where p = a.n
r is the position vector of a point on a plane and n is a normal to the plane
Write n as a vector and p as a scalar
Cartesian equation of a plane
Set r = (x,y,z) and rearrange to equal 0
anx + bny + cnz - p = 0
Angle between a line and a plane
Find the angle between the line and the normal and do 90 - θ
Angle between two planes
Find the angle between the two normals and subtract from 180, take whichever acute answer you get
Skew Lines
Two lines which do not meet
Find the point where two lines intersect
Write as column vectors, solve the first two equations simultaneously for μ and λ then substitute into the third to check
Intersection of a line and a plane
Write the line as a column vector and substitute as r in the plane, solve for λ using the dot product then substitute back in
Shortest distance between two lines
1) Group together for A and B
2) Carry out B-A for AB
3) Take the dot product with each direction and set to 0
4) Solve simultaneously for μ and λ
5) Substitute into AB and take the modulus
Shortest distance between a point and a line
1) Group together each component of the line’s vector
2) Subtract the point from the line
3) Set the dot product of that and the line’s direction to 0
4) Use that to find λ
5) Substitute and take the modulus
Shortest distance between point x and plane r.n = d
|n|
Reflections of points in planes
1) Write a line as r = point + λ(direction of normal)
2) Substitute that as r in the plane equation
3) Solve for λ
4) Substitute 2λ in place of λ in r = point + λ(direction of normal)
Reflecting lines in planes
1) Write the vector equation of the line
2) Substitute as r in the plane equation and find the point of intersection
3) Reflect the point where λ=0 in the plane
4) Find the line which passes through those two points - don’t use λ
Finding a plane that contains a line and a point
Let λ=0 and then -1 to find another 2 points
Write as simultaneous equations equalling 1 and solve
Scale so x,y,z are integers
Write in the correct form
Plane vector to Cartesian
Use x,y,z = a + bλ + cμ
Use the y,z equations to find μ and λ in terms of y and z
Substitute into the x equation
