Vectors and Matrices Flashcards
How can vectors be written?
xi + yj or xi + yj + zk
(x) or (x)
(y) (y)
(z)
ABHow to calculate the magnitude of a vector
sqrt(x^2 + y^2) or sqrt(x^2 + y^2 + z^2)
How to work out a unit vector
(x/magnitude)
(y/magnitude)
(z/magnitude)
Magnitude notation
|(x)| or |xi + yj + zk|
|(y)|
|(z)|
|AB|
How can equation of lines be written?
Position Vector + constant(Direction Vector)
x) (3) (1
(y) = (2) + λ (3)
(z) (4) (4)
How can you write the equation of a line in cartesian form given the vector form?
Form equations for each co-ordinate and re-arrange for λ or the constant. Then equate the equations.
How to find the equation of a line given 2 points?
Points a and b.
Direction = b - a, position = a.
So a + λ(b - a).
How to find if two lines are the same, intersect or parallel (2D)
If direction vectors are multiples of each other then parallel or same.
If they share a point (use a position vector) then same,
If intersect, equate the equations for x and y in both.
Solve simultaneous equations for λ and constant, plug values in for co-ordinates.
How to find if two lines are the same, intersect, skew or parallel (3D)
Same as 2D for parallel or same.
Solve simultaneous equations for λ and other constant, then put into z equation.
If z equation works, then there’s an intersection, so plug values back in.
If it doesn’t work, lines are skew.
Name the 2 dot product formulae for lines with direction vectors (a) (x)
(b) and (y)
(c) (z)
Calculating the dot product: ax + by + cz
cos(θ) = a . b/(|a||b|)
θ = angle between 2 vectors where both tips are touching, or both tails are touching of the vectors.
What are 2 lines in relation to each other if their dot product is 0?
Perpendicular.
How do you find the shortest distance between a point and a line?
Any point on line can be written using equation of line, so can write co-ordinates of point as (a + λd, b + λe, c + λf)
Find equation of line between that and point, take the direction vector.
Equation of line between that and position vector = direction vector.
Their dot product = 0, so solve for λ, then substitute to find point, then solve for magnitude.
How to find the shortest distance between 2 points
Points (a, b, c) and (x, y, z)
sqrt((x - a)^2 + (y - b)^2 + (z - c)^2).
What should you always use in a vector question?
Diagram drawn as 2D (even if 3D).
How can you do dot product calculations on a calculator?
Optn Mat/Vct F2 F6 x 2 DotP( F2 Exit x 2 Math (F4) Mat/Vct F1 Put vector 1 in and comma in the middle, then vector 2
Adding Matrices
Add corresponding elements from each matrix, matrices must have same dimensions.
Multiplying
Must have same number of elements in each row in one as columns in the other.
Multiply the first term in the row of matrix 1 by the 1st element of the 1st column of matrix 2. Then do the 2nd element in the row and second element in the column, and add all the products. Then put this sum in the corresponding element in the result matrix, so if it is the top row and middle column, top middle element.
Basic Matrix rules
AB != BA A(BC) = (AB)C A(B + C) = AB + AC A0 = 0 AI = A = IA A - A = 0
What are the names of the 2 matrix rules:
A(BC) = (AB)C
A(B + C) = AB + AC
A(BC) = (AB)C - Associative
A(B + C) = AB + AC - Distributive
How do you find the transformation matrix from the transformation?
Consider where (1, 0) and (0, 1) would go, and the co-ordinates are your left and right columns. Or, (1, 0, 0), (0, 1, 0) and (0, 0, 1).
How do you find the transformation from the transformation matrix?
Check to see if the matrix is a rotation matrix or an enlargement (whether parallel to 1 axis, for all axes or on a plane), and then if neither, see if it’s a reflection.
What’s the 2D rotation matrix?
(cosθ -sinθ)
sinθ cosθ
What are the matrices for reflection in the planes x = 0, y = 0 and z = 0?
(x 0 0)
(0 y 0)
(0 0 z)
Where if plane x, x = -1, and the others are 1.
Where if plane y, y = -1, and the others are 1.
Where if plane z, z = -1, and the others are 1.
What are the matrices for rotation in 3D around the axes x, y and z respectively?
(1 0 0 )
(0 cosθ -sinθ)
(0 sinθ cosθ)
(cosθ 0 sinθ)
(0 1 0 )
(-sinθ 0 cosθ)
(cosθ -sinθ 0)
(sinθ cosθ 0)
(0 0 1)
How can you multiply or add matrices on a calculator?
Math (F4)
Mat/Vct (F2)
Select vector or choose vector with set dimensions, then put in operator (x, +, -, whatever) and then select other vector.
If you know a combination transformation matrix is a combination of a rotation followed by an enlargement, how can you find the magnitude of the enlargement and θ, the angle of rotation?
k = length of line = square root of the sum of the squares of all the elements in the columns.
(a b) sqrt(a^2 + b^2)
(c d) sqrt (c^2 + d^2)
Divide matrix by k and solve for θ.
