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• Vectors

The equation of a line can be represented by vectors in 3D. The line that passes through A and R can be written as: r = a + λb where a is the position vector of a known point on the line b is the direction vector of the line (a vector parallel to the line) r is the position vector of any arbitrary point on the line λ is a scalar parameter If you need to find the equation of a line from just two points, C and D, and you do not know the direction vector, use the fact that the vector between the two points (d - c) is the direction vector of the line: The equation of this line is given as r = c + λ(d-c) Equations of 3D line in Cartesian Form If you need to find the equation of a straight three dimensional line in Cartesian form (in terms of x, y and x), convert using the vectors a and b from the vector equation r = a + λb. Equations of a Plane in 3D The vector equation of a plane is given as: r = a + λb + μc where a is the position vector of a known point, A, in the plane b and c are non-parallel, non-zero vectors in the plane r is the position vector of any arbitrary point, R, in the plane λ and μ are scalar parameters Normal Vector The normal vector of a plane is used to describe the direction of the plane. It is the vector that is exactly perpendicular to the plane. Equation of a 3D plane in Cartesian Form You can use the normal vector of a plane to write a Cartesian equation describing the plane. Where the normal vector, n = ai + bj + ck, the Cartesian equation of the plane is given as: ax + by + cz = d a, b, c and d are all constants Scalar Product (a.k.a "Dot Product") The scalar product, a . b, is given by the magnitude of the two vectors a and b, and by the angle between them, θ: a . b = |a| |b| cos(θ) If θ = 90° (the two vectors are perpendicular), then the scalar product will equal zero. If a and b are parallel, a . b = |a| |b| If the two vectors are identical (both are a), the scalar product will equal |a|² In Cartesian Form Scalar Product Equation of a Plane The equation of a plane can also be written as the scalar product of the normal vector to the plane and the position vector of any arbitrary point on the plane: r . n = k r is the position vector of any arbitrary point on the plane n is the normal vector of the plane k is a scalar constant for the plane, where k = a . n for a specific point in the plane with position vector a Angles between Lines & Planes Angle Between Two Lines The acute angle between two intersecting lines, with direction vectors a and b, is given as above. |a| |b| represents the magnitude of each vector - use 3D Pythagoras for this. To find the obtuse angle, subtract the acute angle from 180°. Angle Between a Line and a Plane The acute angle between the line r = a + λb and the plane r . n = k is given as above. again, |b| |n| represents the magnitude of each vector - use 3D Pythagoras for this. Angle Between Two Planes The acute angle between the plane r . n₁ = k₁ and the plane r . n₂ = k₂ is given as above. Points of Intersection When you have the vector equations of two lines, you can see if they intersect or not. If the two lines are parallel, they never intersect. Write out the two equations as column vectors Write each row of the column vectors as simultaneous equations involving λ and μ Solve two of the three equations for λ and μ See if these values are consistent with the third equation There are a number of possible outcomes: If all three equations are consistent with the values of λ and μ, the lines intersect at those values of λ and μ If two of the three equations have solutions, but the third does not match, the lines do not intersect. If none of the three equations can be solved simultaneously, the lines do not intersect. If the two lines are neither parallel nor intersect, they are skew Finding Perpendiculars Shortest Perpendicular Distance Between a Point and a Line Shortest Perpendicular Distance Between Two Non-Intersecting Lines Shortest Perpendicular Distance Between a Point and a Plane Unlike the above two examples, there is a fixed equation that can be used to find the perpendicular distance between a point and a plane: All you have to do is substitute in the values.

• Further Integration

• Polar Coordinates

The Cartesian system in two-dimensions models points in terms of x and y. The polar system, however, models a point as a distance form the pole, r (generally the origin) at a certain angle from the initial line, θ (typically the positive horizontal axis). Yes, this is like the modulus-argument form of complex numbers and Argand diagrams. From the diagram, we can derive equations to convert between polar and Cartesian systems: r cos(θ) = x r sin(θ) = y Where θ is given by: θ = arctan(y/x) And r is defined using Pythagoras' theorem: r² = x² + y² Sketching Polar Curves To sketch a polar curve, use a graphical calculator or draw u a table of values for regular intervals of θ. This can be done quickly using the table function on the CASIO ClassWiz fx-991EX, and we recommend using π/6 as an interval. The curve in this example is known as a cardioid, due to its dimple. This is common for equations in the form r = a(p+qcos(θ)), but only if q ≤ p < 2q. When p ≥ 2q, there is no dimple, making it more egg-shaped: Areas Enclosed by Polar Curves The area of a sector of a polar curve can be calculated using integration. However, simply integrating r will not work. Instead: Of course, you an also calculate areas between polar curves. To do this, you need to find the angle at which they intersect. Tangents to Polar Curves To find tangents to a polar curve, you need to convert it into Cartesian form (one equation for x and one for y, both in terms of θ), using the formula at the top of this notes sheet. Then, you can differentiate parametrically. Standard results are:

• Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions, but are defined in terms of exponentials. There are three fundamental hyperbolic functions: sinh, cosh and tanh: Similarly, the reciprocal of each function exists: Hyperbolic Graphs For any value of x, sinh(-x) = -sinh(x) y = sinh(x) has no asymptotes For any value of x, cosh(-x) = -cosh(x) y = cosh(x) never goes below y=1 y = tanh(x) has asymptotes at y = ±1, and always stays between these Inverse Hyperbolic Functions Just like sin, cos and tan, the hyperbolic functions have inverses, arcsinh, arcosh and artanh: The graphs of these are their respective reflections in the line y=x: Hyperbolic Identities & Equations The same identities exist for hyperbolic functions as they do for trigonometric functions: sinh(A ± B) ≡ sinh(A) cosh(B) ± cosh(A) sinh(B) cosh(A ± B) ≡ cosh(A) cosh(B) ∓ sinh(A) sinh(B) Equations with a sinh² in them, however, are different: cosh²(x) - sinh²(x) ≡ 1 Note that here, the sinh² is negative (the trigonometric identity is sin² + cos² ≡ 1). This is known as Osborn's rule: According to Osborn's rule, when using trigonometric identities as hyperbolic identities, any sinh² must be multiplied by -1. Differentiating Hyperbolic Functions This is very similar to trigonometric functions: Note that the derivative of cosh(x) is positive sinh(x), not negative. The inverse functions differentiate as such: Integrating Hyperbolic Functions Simply the reverse of differentiation, but remember the " +c " and that the signs are different: The inverse functions can also be integrated: These standard results for when the equation you need to integrate does not have either (x²+1) or (x²-1) in the root in the denominator:

• Matrices & Linear Transformations

A matrix is a system of elements within a pair of brackets. The size of the matrix is given as the number of rows and columns in it. A square matrix is one where there is an equal number of rows and columns A zero matrix is one in which all the elements are zero An identity matrix is a square matrix where the all the values on the diagonal from top left to bottom right are 1, and all other values are zero. This is noted as capital i, I with a subscript number afterwards to show its size. Working with Matrices Adding and Subtracting To add or subtract matrices, you add or subtract the corresponding elements in each matrix. You can only add or subtract matrices of the same size Multiplying by a Scalar To multiple a vector by a scalar, simply multiple each element in the vector by the scalar. This can be factorised out, too. Multiplying One Vector by Another Matrix multiplication is only possible if the number of columns in the first matrix equals the number of rows in the second matrix. The result of the multiplication is known as the product matrix, and will have the same number of rows as the first matrix and the same number of columns as the second matrix. Order of matrix multiplication matters For two matrices, A and B: In general, AB ≠ BA If AB exists, it does not mean BA exists To find the product of two matrices, multiply the elements in each row of the first matrix by the elements in each column of the second matrix Determinants The determinant of a square matrix is a scalar value that represents the matrix. It is only possible for square matrices. The determinant of a matrix, M, is written as det M, or as |M| A singular matrix has a determinant of zero. A non-singular matrix has a determinant that is not zero. To find the determinant of a 2x2 matrix: To find the determinant of a 3x3 matrix: Inverse Matrices The inverse of a matrix M is the matrix M¯¹ M M¯¹ = M¯¹ M = 1 Finding the Inverse of a 2x2 Matrix Find the determinant Write out the matrix where a and d are swapped, and both b and c are multiplied by -1 Multiply by one over the determinant If A and B are non-singular matrices, (AB)¯¹ = B¯¹A¯¹ Finding the Inverse of a 3x3 Matrix Find the determinant Replace each element with it matrix of minors & find the determinants Form the matrix of co-factors (switch every other sign) Find the transpose of the matrix of co-factors by switching rows and columns Multiply this by one over the determinant Since this is such a complicated thing to do, unless specifically needed, use the calculator function. On the CASIO ClassWiz fx-991EX: Click MODE Click 4: Matrix Click 1: Define Matrix A Click 3 twice to set size of matrix Fill in Matrix Click AC Click OPTN CLICK 3: Matrix A Click x¯¹ Click = Matrix Systems & Geometry It is possible to use matrices and their inverses to easily solve simultaneous equations. This example is consistent, meaning there is at least one set of values that satisfies all three equations simultaneously. The determinant is not zero, making the matrix non-singular. This is especially useful for 3D geometry, and the equations of planes: Linear Transformations A linear transformations are transformations with specific properties: they are made up of linear x, y and z terms only they have no non-variable terms they map the origin onto itself they can be represented by matrices Points and lines that do not change in a transformation are called invariant points/lines The origin is always an invariant point. Every point on an invariant line is an invariant point. Reflections The line of reflection is always an invariant line. In two-dimensions, there are standard reflections for the coordinate axes and y=±x. In three-dimensions, reflections ca happen in planes: For a linear transformation with matrix M, if the determinant is negative, the shape has been reflected Rotations For rotations about the origin, the only invariant point is the origin. In 3D, rotations can be about an entire axis: Enlargements & Stretches If a = b, the transformation is an enlargement of scale factor a. For a stretch that is only in the x-direction, the y-axis is an invariant line. For a stretch that is only in the y-direction, the x-axis is an invariant line. For stretches in both directions, there are no invariant lines and the only invariant point is the origin. For a linear transformation with matrix M, the determinant is the area scale factor. Successive & Inverse Transformations It is possible to apply multiple transformations, one after the other. In this instance, order of matrix multiplication is particularly important. For two linear transformations, represented by the matrices P and Q respectivley: PQ represents the transformation Q followed by P A linear transformation with matrix A can be undone with the transformation represented by the inverse matrix, A¯¹.

• Differential Equations

In single maths, first-order differential equations are the only ones looked at, and are solved by separating the variables: When dy/dx = f(x) g(y), you can say ∫ 1/g(y) dy = ∫ f(x) dx Move all the y terms to the left where the dy is Move all the x terms to the right, including the dx This allows you to integrate each side with respect to the variable on that side to solve the equation. You only need to add the ' +c ' to one side. Just like when integrating an indefinite function, the initial result is a general solution and could be anywhere along the y-axis (see section on indefinite integral functions above). To fond the particular solution, you need to know a coordinate point on the curve - sometimes this is called a boundary condition. Integrating Factor This is not covered in single maths, but is an important method of solving first-order differential equations where x and terms are multiplied by one another in one of the terms: Rearrange to be in the form dy/dx + P(x)y = Q(x) Find the integrating factor using the formula above Multiply the dy/dx by the integrating factor and by y Multiply the right hand side by the integrating factor & simplify (if you can) The middle term, P(x)y disappears Move the dx to the right hand side and integrate this side Rearrange to make y the subject - this is the general solution Find the particular solution by substituting in boundary conditions (if there are any) At further stage, we also look at second-order differential equations - these come in two types: homogenous and non-homogenous. Second-Order Homogeneous Differential Equations Second-order homogeneous differential equations can be solved when in the form: Homogeneous means it equals zero The solution in terms of y will have four constants that need to be found: λ and μ, which are found by solving the auxiliary equation A and B, which can only be found if you have boundary conditions The auxiliary equation is am² + bm + c = 0, and its solutions are λ and μ a, b, and c are the coefficients in the second-order homogenous differential equation above. The format of the solution in terms of y depends on the auxiliary equation: Second-Order Non-Homogeneous Differential Equations Second-order homogeneous differential equations can be solved when in the form: The left hand side of this equation is known as the corresponding homogeneous equation and its solution is called the complementary function. The right hand side of this equation is a function of x and its solution is called the particular integral. Non-homogeneous means the corresponding homogeneous equation equals a function of x To solve a second-order non-homogeneous differential equation, first solve the corresponding homogeneous equation as normal (see above), and then find the particular integral: Take the correct standard function Find dy/dx and d²y/dx² Substitute these into the initial equation Solve to find the constants in the standard function The form of the standard function depends on the form of f(x): When the particular integral is already in the complementary function, you need to multiply the particular integral by x For example, if f(x) is in the exponential form where k is the same as one of the roots in the auxiliary equation, multiply the particular integral by x. The complete solution to the non-homogenous second-order differential equation is given as the sum of the complementary function and the particular integral: Solution = complementary function + particular integral Note that in this example, the exponent in the particular integral is -x. If it were -2x or -3x, it would be the same as the solutions to the auxiliary equations, so we would have to multiply the particular integral by x. Simple Harmonic Motion Simple harmonic motion (SHM) is when a particle always accelerates to a fixed central point (or origin) on its line of motion. The acceleration is proportional to the displacement of the particle from the origin. It may be useful to check out the notes sheet on SHM from A-Level Physics before reading on. SHM can be modelled with second order differential equations, because: x is displacement v is velocity, displacement with respect to time a is acceleration, velocity with respect to time Therefore: Often, we represent this with dot notation, where the dot above a variable means that that variable is differentiated with respect to time. Since the acceleration is always proportional to displacement, and always directed to the origin, O, there must be a negative constant of proportionality. This is called the angular velocity, and is represented by ω². Terminology Amplitude is the maximum displacement of the particle from the origin Period is the time taken to make one complete oscillation: this is from the origin out once, through it again to the other side, and then back to the origin. If an object is oscillating, this generally means it is moving with SHM R-Addition Formula When x is given in the form A cos(ωt) + B sin(ωt), it can be written as r sin(ωt - α). Here, r is the amplitude of the SHM, and is found as: r² = a² + b² tan(α) = A/B Damped and Forced Harmonic Motion Harmonic motion is only simple if there are no external forces slowing it down or speeding it up. Damping If there is an external force slowing down the motion of the particle, it is known as a dampening force. This is modelled by second-order homogeneous differential equations with a dx/dt term in the middle: There are three separate cases, depending on the discriminant of the auxiliary equation: Forced Motion If energy is being put into the system, and the oscillating object is driven by an external force, it can be modelled with second-order non-homogeneous differential equations: Questions relating to these contexts are solved exactly as any second-order differential equations should be solved. Coupled First-Order Simultaneous Differential Equations Often in nature, the rate of change of one thing is dependent on the rate of change of another thing. A common example is how the population size of predator affects that of the prey, and vice versa: If the population of rabbit is large, the population of foxes increases If the population of foxes is large, the population of rabbits decreases In such an example, there are two variables (one for the size of each population), and so there are two differential equations to represent how each changes with respect to time. Naturally, these can be solved simultaneously. If r is the number of rabbits at time t, and f is the number of foxes at time t, we can write: If both f(t) and g(t) equal zero, the system is homogeneous. To solve coupled first-order differential equations, you need to eliminate one of the variables.

• Modelling in Mechanics

The real world of mechanics is incredibly complicated, with hundreds of different factors affecting motion and stability. This would be near impossible to calculate at this stage, so to simplify it we model objects and scenarios in a number of different ways. Modelling Assumptions Particle Has negligible dimensions Mass acts about a single point Rotational forces and air resistance can be ignored Rod The diameter is negligible (so no thickness), only length counts Mass acts along a line It is rigid, so does not bend or buckle Lamina An object with only area, thickness is negligible (like a sheet of paper) Mass acts across the flat surface Uniform Body The Mass is distributed evenly throughout the object The mass is modelled to act through one point, the centre of mass, in the geometrical middle of the body Light Object The mass of the object is negligible Often used for strings or pulleys Allows us to model the tension on each end of a string as equal Inextensible String A string that does not stretch when a force is applied Allows us to model the acceleration of two connected particles as the same Smooth Surface There is no friction between the surface and any object on it Rough Surface There is friction between the surface and any object on it Wire A rigid length of metal with negligible thickness (only modelled with length) Smooth and Light Pulley The pulley has no mass No friction in the pulley, so tension is the same either side of it Bead A particle with a hole in it for string/wire to be threaded through It can move freely along the wire or string Peg An object from which a body can be hung or rested Modelled as dimensionless and fixed Can be either rough or smooth Air Resistance Resistance forces due to motion through air Almost always modelled as negligible Gravity The force of attraction between all objects Acceleration due to gravity is given by g, which is 9.8 m/s² unless otherwise stated Assume objects are always attracted to the ground (earth) Gravity acts uniformly and downwards Quantities & Units Quantities can be either scalar or vector. Scalar quantities have only magnitude Vector quantities have both magnitude and direction SI Base Units Quantity Unit Symbol Mass kilogram kg Distance metre m Time seconds s Distance is scalar - the vector version of it is displacement Derived Units Quantity Unit Symbol Velocity metres per second m/s Acceleration metres per second² m/s² Force Newtons N Velocity is the vector form of speed, as velocity has direction as well as magnitude Weight is not the same as mass. Weight is a force, measured in Newtons, N. It is given by: W = mg Weight = mass x gravity Representing Forces Weight acts down from an object's centre of gravity. Other forces can act in any direction, and so to avoid confusion it is best to draw a force diagram: The length of arrow should represent the magnitude of the force. Single headed arrows represent forces. If a system is in equilibrium - meaning there is no net resultant force and it is motionless or at constant velocity - then all the force arrows will make a closed shape. The example above would give a triangle.

• Acceleration, Projectiles & Kinematics

Velocity is the rate of change of an object's position, and therefore has direction. This makes it a vector quantity, unlike speed which is scalar (magnitude but no direction). velocity = distance / time The units of velocity are m/s Acceleration is the rate of change of velocity, and so is the mathematical derivative of this. acceleration = change in velocity / change in time The units of acceleration are m/s² Motion Graphs We can plot motion on two main types of graph - it is important to know the properties of each. Displacement-Time Graphs The Gradient is the velocity - draw a tangent to find the instantaneous velocity Horizontal line represents zero velocity Velocity-Time Graphs The Gradient is the acceleration The Area beneath the graph is the displacement Constant Acceleration When acceleration is constant (e.g. free fall when we ignore air resistance), we can use SUVAT equations to work out the variables: s is for displacement u is for initial velocity v is for final velocity a is for acceleration t is for time Vertical Motion due to Gravity The gravitational force of the earth causes all objects to accelerate towards the ground, its surface. Ignoring air resistance, the acceleration is constant, and given a g: g = 9.8 m/s² This is independent of the mass, shape or velocity of the object. It is vital to set a positive direction of motion for each question In the example above, we set up as the positive direction, so our values for a and v were negative because the ball is going down. Constant Acceleration with Vectors Additionally, we can express motion using vectors: r = r₀ + v t r is the position vector of the moving object r₀ is the initial position vector v is the velocity vector Four of the five SUVAT equations have vector equivalents: v = u + at v = u + a t s = ut + ½at² s = u t + ½ a t² + r₀ s = vt - ½at² s = v t - ½ a t² + r₀ s = ½(u+v)t s = ½( u + v )t + r₀ v² = u² + 2as has no vector equivalent Projectile Motion When we model a projectile, we ignore air resistance. This means that: Horizontal motion of a projectile has constant velocity: a = 0 Vertical motion of a projectile motion is modelled as gravitational free fall: a = g For horizontal projection, like in the diagram above: Since horizontal velocity is constant, we can use the equation x = vt (were x is horizontal displacement) For vertical velocity, we need to use SUVAT equations, due to the constant acceleration of g = 9.8 m/s² Horizontal and Vertical Components When given the velocity as a vector or at an angle, you must use trigonometry to find the vertical and horizontal components. Then, treat them separately as above (the horizontal component still has constant velocity) For an object projected at velocity U at an angle of θ to the x-axis: the horizontal component of the velocity is given as U cos(θ) the vertical component of the velocity is given as U sin(θ) The vertical component is decelerating at -9.8 m/s² as it rises, and accelerates at 9.8 m/s² as it falls The projectile reaches its maximum height when the vertical component of the velocity is zero. Applying SUVAT to the vertical component of the projectile gives us a few standard and useful equations: Projectile motion can also be plotted and calculated with in vectors. Variable Acceleration As you can see from the velocity-time graph, varying acceleration produces curves: This is because the gradient of a velocity-time graph is the acceleration, and so if the acceleration changes with time, so must the gradient. Kinematics Velocity is the rate of change of displacement. Acceleration is the rate of change of velocity. Therefore: This means we can differentiate and integrate equations for motion: Integrating and Differentiating Vectors Variable acceleration can also be expressed in vector form. Therefore, you need to be able to differentiate and integrate vector equations. Often, dot notation is used to quickly represent differentiation with respect to time: To integrate vectors: In both differentiation and integration of vectors, you must do one term at a time See Notes Sheets on differentiation and integration in Pure Maths if you need a recap.

• Forces, Friction & Motion

As seen in the Notes Sheet on Modelling in Mechanics, forces on an object can be expressed with a force diagram: Newton's 1st Law: On this diagram, you can see that T, the tension force pulling the object to the right, is equal to F, the friction between the the object and the surface - friction is the force that opposes motion. This means that there is no resultant force to the left nor right, so the object remains motionless (it is in equilibrium). This is Newton's first law: An object will stay at a constant velocity, or motionless, unless it experiences a resultant force. Newton's 2nd Law: Here, the surface is smooth (no friction). This means the object experiences a resultant force, T, to the right. According to Newton's second law, the resultant force on an object is directly proportional to the rate of change of momentum of the object, and acts in the same direction. This is commonly expressed as: F = ma Resultant Force = Mass x Acceleration (in direction of resultant force) Newton's 3rd Law: In this example, the object sits at rest on the surface with no horizontal forces acting on it. However, its weight, W, (mass x g) is a force acting down. If this were the only force, you would expect the object to accelerate down through the surface but it does not. This means there is an equal but opposite force acting upwards: the normal reaction, R. For every action, there is an equal but opposite reaction (equal in magnitude and type of force). Forces in Two Dimensions Often, forces will not be exactly horizontal or vertical. Instead, they may be given as vectors or at angles. Sometimes you will need to find the resultant from two forces at right angles to one another, but more often you will be given the resultant and need to calculate the components. Use Pythagoras to do this. When given the resultant at an angle, use trigonometry to find the components. Moments Moments are the turning forces acting on an object, around a pivot. M = Fx moment = force × perpendicular distance from pivot The units of moments are Nm If there is a resultant moment in either direction (clockwise or anti-clockwise), then the object will rotate in that direction about its pivot. When an object is in equilibrium, there is neither a net force nor a net moment in any direction. The Principle of moments can be used for this: the sum of all clockwise moments must equal the sum of all anticlockwise moments for an object to be in equilibrium. When we model a rod or object as uniform, it means we assume all the weight is evenly distributed throughout, and only acts about the centre. Tilting The point of tilting is the point at which a rigid body is about to pivot. When a rigid body is at the titling point, the reaction at any other support is zero. Friction Friction is the force that opposes motion. No materials are perfectly smooth, but have a degree of roughness: this is defined by the coefficient of friction, μ, which is a fixed property of a surface in certain conditions (conditions matter, because if a surface is wet, its coefficient of friction will be lower). An object will only move on a surface if the force applied to it is greater than the maximum friction, F(max): this value is knows as the limiting value. The limiting value, F(max), depends on both the normal reaction force, R, and the coefficient of friction, μ: F(max) = μR Limiting Value = maximum friction = μR Static Rigid Bodies A rigid body is something like a rod or a ladder, leaning against a wall. If it is static, it is in equilibrium. For a rigid body to be in equilibrium: It must be stationary There must be no resultant force in any direction There must be no resultant moment A common example is a ladder leaning against a wall, where there is friction on the ground: Connected Particles & Pulleys When the particles are moving along the same straight line, they can be modelled as one particle with a combined mass. When the particles are moving separately, for example in a pulley system, they can be treated as separate particles. In both instances, when the string is light and inextensible, the acceleration in each particle will be the same. A common question will consist of two connected particles, one on an inclined plane, the other hanging off a pulley:

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