Maths Flashcards
Why does the vector product of two vectors in a plane give us the coefficients of the variables in the standard form equation of that plane?
Because if you take the dot product of ANY vector [xi, yj, zk] in the plane and the vector then the answer is 0. See Feb 7 lecture if you don’t understand this anymore
How do you find the determinant of matrix A, det(A)?
If matrix A is
|a b|
|c d|
Then det(A) also written |A| is ad-bc
How do you find the inverse of matrix A (the matrix that you multiply A by to get the identity matrix I)?
A^-1 = 1/det(A) *
|d -b|
|-c a|
What is a linear transformation?
Linear transformations do not depend on the order of scalar multiplication of vectors or addition of vectors. E.g. Rotation is a linear transformation because you can add two vectors then rotate them then multiply them by a scalar or you can rotate them then add them then multiply them and you’ll get the same thing (this can be done in any order).
What are the steps for performing a linear transformation with matrices?
1) do the operation on the i and j (and k for 3D) vectors. Let it, jt, and kt be the transformed vectors
2) make a matrix with the first column it, the second jt, and the third kt
3) to get Vt for any vector V, just do Mt * V where Mt is the transformation matrix created in part 2
How many rows and columns does the transformation matrix have?
Since you're doing Mt * [x,y,(z)] it needs as many rows as the vector (the domain) and as many columns as the final vector you want. Since the dimensions of the final matrix are #colMt * #rowVector (which is always 1).
Double angle formulae
Ben
Differentiation and integration
Ace this shit please
Scalar product vs vector product
Scalar product is 0 if they are perpendicular and vector product is 0 when they are parallel
Scalar:
v•w = |v||w|cos(x)
Vector:
v x w = |v||w|sin(x)
What relates v x w and w x v?
v x w = - w x v
How to write the equation of a plane (going through the origin) in standard form from vectors a and b?
Do a x b, you’ll get vector [ABC]
The equation of the plane is:
Ax+By+Cz=0
Intersection of 2 planes in 3D through the origin
Since hey share the origin they must share a line, set them equal to each other (since they’re both equal to zero as they pass through the origin) and you’ll get the equation of a line
Intersection of 3 planes through the origin in 3D
Each pair shares a line, to see if all three share a line or only a point (the origin), isolate y for equation 1, then plug that into the y of equation 2 to isolate z in terms of x. Now you have each variable in terms of x (x=x, z is in terms of x, and y is in terms of x and z (which is in terms of x)). Now plug these xyz in terms of x into equation 3 and the solutions that make it work are the points that lie on all three planes (if it only works with x=0 then only the origin is on all three planes)
What dimensions do you need to be able to multiply two matrices together?
If matrix A is f by G, to have AxB matrix B must be G by h and the final matrix will be f by h
Top right, bottom left
When multiplying two matrices A and B, AxB, the top row of A and rightmost column of B make the top right value of he new matrix, and the bottom row and leftmost column of B make the bottom left value of the new matrix
