TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > MFP4 AQA Mathematics Current Spec Alevel
Notes for FP4, Fun stuff :|
- 2 matrices can be added or subtracted if they have the same order. To add or subtract, just add or subtract the corresponding elements.
- Any matrix can be multiplied by a scalar. Just multiply each element by the scalar.
- Matrices are multiplied by multiplying the elements in a row of the first matrix by the elements in a column of the second matrix, and adding the results.
- Commutativity of addition : For any 2 matrices which can be added, i.e. have the same order, A + B = B + A
- Non commutativity of multiplication : It cannot be assumed that BA = AB
- Distributive law: A(B+C) = AB + AC (U+V)W = UW + VW
- Associative law : A(BC) = (AB)C
- A + 0 = A, 0B = 0, C0 = 0
- AI = A, IB = B
- M = AB and N = BC Explain why AN = MC
Multiply the 2nd equation by A… AN = ABC as AB = M AN = MC
- The transpose of a matrix M is obtained by interchanging the rows and columns of M. I.e. reflecting in the leading diagonal. The transpose of M is denoted by MT.
- (AB)T = BTAT
- Identity : all points unchanged
- BA is the transformation A then the transformation B
- Any linear transformation has the properties:
T(λa) = λT(a) T(a + b) = T(a) + T(b)
- cos2θ = cos2θ – sin2θ sin2θ = 2sinθcosθ
- The scalar product of 2 vectors a and b : a.b = |a||b|cosθ
- The vector product of 2 vectors a and b : a x b = |a||b|sinθn
- i x i = j x j = k x k = 0
- i x j = k j x k = i k x i = j
- j x i = -k k x j = -i i x k = -j
- a x b = 0 a = 0 or b = 0 or a and b are parallel.
- For any 2 vectors a and b, b x a = -a x b
- For k and m any scalars, ka x mb = km a x b
- The area of a triangle is 0.5|axb|
- The area of a parallelogram is |axb|
- a.bxc means a.(bxc) This is a scalar quantity – a scalar triple product. a x (bxc) is a vector quantity – a vector triple product.
- a.bxc = b.cxa = c.axb
- If three adjacent edges of a parallelepiped are represented by the vectors a, b and c then the volume of the parallelepiped is |a.bxc|
- a.bxc = axb.c
- The vector product is distributive over addition:
(a + b) x c = a x c + b x c
- Prove that : a x (b+c) = a x b + a x c
r. a x (b+c) = r x a . (b+c) [interchange dot and cross] = r x a.b + r x a.c [distributivity of dot] = r.axb + r.axc [interchange dot and cross]
r. a x (b+c) = r.axb + r.axc r. a x (b+c) - r.axb - r.axc = 0
r. (ax(b+c) – axb – axc) = 0
- |M| is negative if the transformation represented by M is a reflection or involves a reflection.
- The determinant of M is defined to be a.bxc and is written det(M) or |M|
- Determinant of a 2x2 = (ad – bc)
- The determinant is the scale factor of the enlargement of an area if 2x2 and a volume if 3x3
- |M| = |MT|
- Adding or subtracting any multiple of a row/column to another row/column doesn’t change the determinant.
- Swapping 2 rows/columns changes the sign of the determinant.
- Multiplying a row/column of a matrix by k multiplies the determinant by the k
- To factorise a determinant use row/column operations to get a row/column of elements with a common factor.
- |AB| = |A||B|
- Volume of a cuboid : |a.bxc| Volume of a parallelepiped: |a.bxc|
- Volume of a pyramid : 1/3|a.bxc| Volume of a tetrahedron : 1/6|a.bxc|
- Volume of a triangular prism : 1/2|a.bxc|
- a, b and c are coplanar |a.bxc| = 0
- The equation of a line through a and parallel to b can be expressed:
- parametric : r = a + tb
- vector product : (r-a) x b = 0
- Direction cosines: when a line is in Cartesian form it is simple. Take the vector at the bottom, and use the formula below. E.g. b1 is the i value.
- The direction cosines of a line l,m and n satisfy : l2 + m2 + n2 = 1
- Parametric equation of a plane : r = a + λb + μc
where λ and μ are parameters. r is the position vector of a general point on the plane, a is any specific point on the plane, b and c are 2 non-parallel vectors on the plane.
- r.n = a.n where a is a point on the plane, n is a normal to the plane. r.n = d. Therefore the equation 3x + 2y – 7z = 12 (3, 2, -7) is n and 12 is d.
- Find an equation of the plane through the points: a (1,0,2) , b (3,0,5) and c (-1, 6, 2)
Get 2 vectors : vector 1 = b-a vector 2 = c-b Point : (1, 0, 2)
r = point + λ(vector 1) + μ (vector 2)
- Find an equation of a plane through 2 lines: obtain a vector perpendicular to both lines by crossing the 2 direction vectors. Then do r.n=a.n where a is a point and n is the vector found from the cross product. Can then replace r with (x,y,z)
- Find the line of intersection of 2 planes: find direction of the common line by crossing the 2 n vectors i.e. if 3x + y + 6z = 8 is one plane the (3, 1, 6) needs to be crossed with the other n vector. Then find any common point by setting for example z to zero and doing simultaneous equations, if this doesn’t work then set x or y to zero. Then do r = (common point) + t(n vector)
- Find the point of intersection of a line and a plane: make the line equation into r=(3+t, -2+3t, 6t) then substitute in for x, y and z to get a value for t. Plug this t value back into the line equation to get the point of intersection.
- Angle between line and plane: dot product the direction vector of the line with the n vector to the plane. Obtain a value for . If obtuse, do 180 - = k then do 90 – k. If acute, do 90 - . This because we have found the angle between the perpendicular and the plane not the line and the plane.
- To find the acute angle between 2 planes, just find the acute angle between their perpendiculars. I.e. the dot product.
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- To find the shortest distance between 2 lines:
-Find any vector AB from a point on one line to a point on the other
-Find a vector n perpendicular to both lines
-Calculate |AB.n|
|n|
- The inverse of A, any square matrix, is A-1 such that : AA-1 = A-1A = I
- A square matrix without an inverse is singular. I.e. all matrices with a determinant of 0 are singular.
- Algorithm for the inverse of a 3x3 matrix:
-|M|. If 0, stop. -Matrix of minor determinants -Sign change : + - + , - + - , + - +
-Transpose -Divide by |M|
- If A, B, C … are all matrices of the same size and they all have inverses, so does the product ABC. Also, (ABC…)-1 = …C-1B-1A-1
- When considering a system of 3 equations, ensure that 2 planes aren’t the same or parallel. I.e. they’re the same if they’re multiples of one another; they’re parallel if one equation has the same x y and z coefficients but different constant terms.
- When you do row operations, if it ends up with a row of zeros along the bottom then they’re inconsistent. If the row of zeros equals zero, then the planes meet at a line. If the row of zeros equals k not equal to zero, there are no solutions : prism.
- A-1(Ar) = A-1b => r = A-1b i.e.number number number = A-1(xyz)
- Row operations: equations can be multiplied by any non zero number, and any multiple of a row can be subtracted from or added to another row. Row operations can reduce to 2 equations in 2 unknowns and then to 1 equation in 1 unknown.
- When a vector is a combination of other vectors they are linearly dependant, or when the determinant of the vectors together = 0.
- The invariant points of the transformation of matrix M can be found by solving Mx=x
- The invariant lines of the transformation with matrix M can be found by substituting x and mx or x and mx + c in the same equation of a line.
- If v is an eigenvector then the line r = tv is an invariant line through the origin.
- The eigenvectors of a transformation determine the directions of all the invariant lines that pass through the origin.
- The eigenvectors for a matrix M can be found by solving |M-λI| = 0
- Can show something is an eigenvector by solving Mv=λv
- When you have an eigenvalue, can solve by setting Mx =0 where x is x,y,z matrix
- A diagonal matrix is a square matrix that has all zeros except the leading diagonal. a matrix M which has eigenvalues and eigenvectors can be written as VDV-1 where V is the eigenvectors and D is the eigenvalues. They must be in the right order.
- If M = VDV-1 then Mn = VDnV-1
- Example: The matrix M has eigenvectors v1 and v2 with associated eigenvalues λ1 and λ2 . Find Mv where v = av1 + bv2
Answer: Mv = M (av1 + bv2) {substitute value for v in}
= a(M v1) + b(M v2) {take a and b outside}
= a(λ1 v1) + b(λ2 v2) { Mv = λv so replace Mv with λv}
= λ1(a v1) + λ2 (bv2) {take λ1 and λ2 outside}
