SINGLE MATHEMATICS B : Vectors Summary Notes

Transcription

1 Preprint typeset in JHEP style - HYPER VERSION SINGLE MATHEMATICS B : Vectors Summary Notes Ruth Gregory Abstract: These notes sum up all you need to know about the mathematics of vectors at this stage.

2 Contents 1. Vectors Vectors and Scalars Bases in R Products of vectors The scalar product The vector product Triple products 4 3. Lines and Planes Lines in R Planes in R Polar co-ordinates Plane and cylindrical polars Spherical polars 6 1. Vectors 1.1 Vectors and Scalars A vector is something which has both a magnitude and direction, as distinct from a scalar which is just a number. Vectors are assigned at a point, and can be visualised as arrows with a given length based at that point. They can (at least in a flat space!) also be thought of as displacing you from one point to another. (See figure 1.) Vectors are usually thought of as objects in R 3, i.e. three-dimensional space. This is certainly true for most simple applications, but much of physics has more abstract or complicated vectors for example, relativity is much easier if you view it four dimensionally, and quantum mechanics actually has infinite dimensional Hilbert spaces. Vectors can be added, and multiplied by a scalar (see figure 2). These properties are what defines a vector in abstract. 1

3 a A AB b a b B c BC c b C O Figure 1: An illustration of vectors both as arrows with length and direction, as well as displacements: a is the vector from O to A. The displacements from A to B and B to C are also vectors: e.g. AB = b a. b a+b λc a c b Figure 2: Adding of vectors: to add a and b together, first displace along the vector a, then along the vector b (where you imagine sliding b so that the base of the arrow starts at the arrowhead of a) and shows that addition is commutative. Multiplication by a scalar: the vector c does not change its direction, but simply has its length scaled by λ. 1.2 Bases in R 3 The set {e 1, e 2, e 3 } is a basis for R 3 if any vector v in R 3 can be written uniquely as v = v 1 e 1 + v 2 e 2 + v 3 e 3. The numbers v i are the components of the vector v with respect to the basis {e i }. The standard basis in R 3 is based on the cartesian coordinates {x, y, z}, and is given by the unit vectors in the x, y and z directions. It is denoted by {i,j,k} or {e x,e y,e z }. This gives us three different ways of writing v: as an abstract geometric object, a sum of components and basis vectors, or as an ordered triplet of numbers 2

4 (can be a column or a row) v 1 v = v 1 i + v 2 j + v 3 k = v 2 (1.1) v 3 Addition and multiplication by a scalar are particularly simple in terms of components: v 1 w 1 v 1 + w 1 addition: v + w = v 2 + w 2 = v 2 + w 2 ; v 3 w 3 v 3 + w 3 v 1 λv 1 scalar multiplication: λv = λ v 2 = λv 2. v 3 λv 3 2. Products of vectors 2.1 The scalar product The scalar or dot product is defined as: u 1 v 1 u v = u 2 v 2 = u 1 v 1 + u 2 v 2 + u 3 v 3 = uv cos θ, u 3 v 3 where u and v are the lengths of u and v, and θ is the angle between u and v. Clearly then, the length of v, v or v, is v = v v. The vectors v and w are said to be orthogonal if the angle between them is π/2. The non-trivial vectors v and w are orthogonal if and only if v w = The vector product The vector product or cross product of two vectors is another vector: u 1 v 1 u 2 v 3 u 3 v 2 u v = u 2 u 3 =. v 2 v 3 u 3 v 1 u 1 v 3 u 1 v 2 u 2 v 1 Properties of vector product: w v = v w; v (v w) = w (v w) = 0. u v = u v sinθ ˆn, where ˆn is the unique unit vector normal to both u and v, where {u,v, ˆn} form a right handed triplet. 3

5 2.3 Triple products The scalar triple product of three vectors a, b and c in R 3 is: [a,b,c] = [b,c,a] = [c,a,b]. [a,b,c] = a (b c). For any a, b, c in R 3, [a,b,c] is the determinant of the matrix whose columns are a, b and c. The volume V of a parallelepiped P formed by a, b, and c is V = a (b c) = [a,b,c]. Similarly, the volume of a tetrahedron T is [a,b,c]/6. The Vector triple product: a (b c) = (a c)b (a b)c. 3. Lines and Planes 3.1 Lines in R 3 A line is defined by a point on it, and a direction along it: a 1 r = a + λd = a 2 + λ d 2. (3.1) a 3 d 3 Where a is any point on the line, d is a vector along the line, λ is a parameter running along the line. This is called the parametric form of a line (see figure 3). d 1 d a λ 3.2 Planes in R 3 A plane in R 3 is characterised by a point in it, and either two directions within it, or, Figure 3: The line as a displacement point plus direction vector. equivalently, a normal to the plane. Since the plane is a flat two dimensional surface in a three dimensional space, it can also be characterised by a linear constraint. These give the two main equations for a plane: Constraint equation ax + by + cz = l for fixed real numbers a, b, c, l. 4

6 Using normal vector: x n r = n y = n a z l/a where n is a normal vector and a is a fixed point in the plane, such as Polar co-ordinates 4.1 Plane and cylindrical polars If P is a point in the plane (x, y) then it is often useful to set x = r cos θ y = r sin θ, see figure 4. We can then set up a new vector basis based on these polar coordinates: {e r,e θ } (see figure 4). y e θ er r (x,y) θ x Figure 4: Plane polar coordinates and their associated orthonormal basis. e r = cosθi + sin θj ; e θ = sin θi + cos θj. Since i and j are fixed, the only variable in these equations is θ, so differentiating gives ė r = θe θ ; ė θ = θe r. 5

7 Hence ṙ = ṙe r + r θe θ r = ( r r θ 2 )e r + (r θ + 2ṙ θ)e θ These coordinate generalize to a cylindrical system by adding in the z-direction, and the basis vector e z = k. It seems a bit strange at first to set up a basis at individual points in the plane, but actually this mirrors the definition of vectors in general spaces. In many physical problems you want to know the value of a vector field at various points in space - what is more natural then than to express that vector in terms of some basis at that point? Many physical problems also have rotational symmetry, and it can be easier to express vectors in this new basis. Radial and axial vectors have a very simple form, see figure 5. I B E Figure 5: An example of an axial and radial vector field. On the right, a line charge produces a radial electric field E e r. On the left, a current in a wire produces an axial magnetic field B e θ. 4.2 Spherical polars Spherical polars are coordinates adapted to spherical symmetry in R 3. They are defined via; x r sin θ cosφ r = y = r sin θ sin φ. z r cos θ Once again, r is the distance from the origin, φ is the angle in the {x, y} plane, and θ is the angle r makes to the z-axis. 6

8 We can form a spherical orthonormal basis: e r = sin θ cosφi + sin θ sin φj + cosθk e θ = cosθ cosφi + cosθ sin φj sin θk e φ = sin φi + cosφj. As before, i, j and k are fixed, so only θ and φ can vary. Differentiating gives ė r = θe θ + φsin θe φ ė θ = θe r + φ cosθe φ ė φ = φ(sin θe r + cosθe θ ), and for a particle moving along a path r(t), we have: ṙ = ṙe r + r θe θ + r φsin θe φ r = ( r r θ 2 r sin 2 θ φ 2 )e r + (r θ + 2ṙ θ r φ 2 sin θ cos θ)e θ + (r sin θ φ + 2ṙ φ sin θ + 2r θ φcosθ)e φ. 7