Vectors J.R. Wilson September 28, 2017 This chapter introduces vectors that are used in many areas of physics (needed for classical physics this year). One complication is that a number of different forms of notation exist for vectors so a variety of nomenclature will be used here. References for this chapter: Stroud Part II programme 6 (pages 591 617), Riley Chapter 7 (pages 212 224) 1 Definitions A scalar can be defined by a single number with appropriate units. (eg. speed, length, area, volume, mass, time, temperature, cost). A vector is defined completely by its magnitude (with units) and direction. (eg. velocity, force, acceleration, weight, wind). 2 Representation In cartesian coordinates we would represent a vector by: v = (v x, v y, y z ) Here v x, v y and v z are the components of the vector relating to the (x, y, z) axes in cartesian space. We can also define unit vectors along the (x, y, z) axes, usually called (i, j, k) or (î, ĵ, ˆk). The unit vectors all have unit length i = î = (1, 0, 0) j = ĵ = (0, 1, 0) k = ˆk = (0, 0, 1) 1
i = j = k = 1 and represent a right handed set of basis vectors. Any vector can be written as a linear combination of these unit vectors. If v x has magnitude x, v y has magnitude y, and v z has magnitude z, we can also define the vector as as shown in figure 1. v = xi + yj + zk Figure 1: A (magenta) vector in the three-dimensional Cartesian space. The magnitude, or length, of vector v can be written v = v = r = (x 2 + y 2 + z 2 ). You may also see vector notation of the form OP to indicate a line from point O to point P, which would have magnitude OP or OP. For two vectors to be equal, both their magnitudes and directions must be the same. ie. if a = b then a = b ( a = b ) and the directions must be parallel and in the same sense. Note that OP P O as these vectors are parallel in the opposite sense. 2
3 Addition of Vectors We can add two vectors, A = (ax, a y, a z ) and B = (b x, b y, b z ) together to obtain C = (c x, c y, c z ) = A + B by adding the respective components: C = A + B = ((a x + b x ), (a y + b y ), (a z + b z )) Vector addition is commutative meaning: Vector addition is associative meaning: A + B = B + A ( A + B) + C = A + ( B + C) We can understand this by considering the components of C in figure 2 Figure 2: Addition of vectors A + B = C = B + A Any single vector OC can be replaced by a sum of any number of vectors so long as they form a chain in the vector diagram. 4 Products of Vectors 4.1 Scalar multiplication To multiply (or divide) a vector by a scalar quantity, each component is scaled by that quantity. λ v = (λx, λy, λz) 3
For any vector, v we can define a unit vector, ˆv, which is a vector that points in the same direction but with unit length by dividing through by the vector s magnitude ( v = r). 4.2 Scalar or Dot Product ˆv = v v = ( x r, y r, z r ) The scalar product of two vectors gives a scalar value v. u = s which corresponds to the sum of the products of the corresponding components of the two vectors: v. u = (v x, v y, v z ).(u x, u y, u z ) = v x u x + v y u y + v z u z = v i u i Geometrically, it corresponds to the product of the moduli of the two vectors and the cosine of the angle between them: v. u = v u cos θ x,y,z where θ is the angle between the two vectors that can range between 0 and π. figure 3. See Figure 3: Scalar Product Basically, the scalar product tells us how alike are two vectors, or also how much of one is in the other. 4
We can use a scalar product with the unit vectors to project a vector along a given axis. eg. Multiplying v by the î unit vector, we obtain the component of v along the x-axis. v.î = v x v.ĵ = v y v.ˆk = v z The scalar product of a vector with itself is the square of the modulus: Scalar products are commutative: Scalar products are associatative: v. v = v 2 x + v 2 y + v 2 z = v 2 v. u = u. v ( v. u). w = u.( v. w) Scalar products are distributive over addition: 4.3 Vector or Cross Product ( v + u). w = u. w + v. w The vector product(or cross product) corresponds to a new vector that is perpendicular to both the original vectors and therefore normal to the plane containing them. The three vectors u, v and u v form a right-handed set as shown in figure 4 such that the right handed rule can be applied to determine the direction of the product. The index finger can be used to represent the first vector, A the middle finger represents the second, B such that the angle θ turns from the index to middle finger ( A to B), and the product A B is represented by the thumb. For the product B A you need to turn your hand so that the index finger now represents B and the middle finger represents A - not the thumb, B A, points in the opposite direction. The magnitude of the resulting vector can be obtained by the geometrical definition of the vector product: w = u v = u v sin θ where θ is the angle between the two vectors. The geometrical interpretation can be seen in figure 5. The magnitude of the vector product gives the area of the parallelogram formed by the two vectors, while the direction is normal to the surface of the parallelogram. 5
Figure 4: Vector Cross Product Direction from the right hand rule. Figure 5: Vector Cross Product Magnitude 6
The vector resulting from the vector product is defined as a pseudo-vector or an axial-vector: this means that it transforms like a vector under a rotation, but it changes sign under a reflection. In physics, there are a number of these pseudovectors, like for example the magnetic field B and the angular momentum L. Vector products do not commutate: Vector products are not associative: u v = v u ( u v) w u ( v w) if we multiply first u and v and then multiply by w we get a different result to multiplying first v and w and then multiplying by u Vector products are distributive over addition: w ( u + v) = w u + w v The cross product of two identical vectors is zero: Applied to the basis unit vectors: and v v = v v sin(0) = 0 î î = ĵ ĵ = ˆk ˆk = 0 î ĵ = ˆk = ĵ î ĵ ˆk = î = ˆk ĵ ˆk î = ĵ = î ˆk Now that we know the properties of the cross product and how the basis unit vectors behave under cross product, we can calculate the cross product between two generic vectors: u v = (u x î + u y ĵ + u zˆk) (vx î + v y ĵ + v zˆk) = u x v x (î î) + u x v y (î ĵ) + u x v z (î ˆk) +u y v x (ĵ î) + u y v y (ĵ ĵ) + u y v z (ĵ ˆk) +u z v x (ˆk î) + u z v y (ˆk ĵ) + u z v z (ˆk ˆk) = u x v y (ˆk) + u x v z ( ĵ) +u y v x ( ˆk) + u y v z (î) +u z v x (ĵ) + u z v y ( î) = (u y v z u z v y )î + (u z v x u x v z )ĵ + (u x v y u y v x )ˆk 7
This can also be represented by the matrix definition: u v = (u x, u y, u z ) (v x, v y, v z ) î ĵ ˆk = u x u y u z v x v y v z = (u y v z u z v y )î + (u z v x u x v z )ĵ + (u x v y u y v x )ˆk The last line here is the determinant calculation for a 3 3 matrix (see figure 6 for a reminder of matrix determinants). Note that the î and ˆk terms follow the cyclic order x y z whereas the ĵ term ordering accounts for the negative sign in the determinant. Figure 6: The determinant of a 3 3 matrix. 5 Angle between vectors The scalar product allows us to determine the angle between two vectors. Rearranging the earlier geometric definition of the product we get: cos θ = v. u v u = 1 v u v i u i 6 Direction Cosines Direction cosines define the angle that a vector makes with the axes of reference. Another way to think of this is to view them as the corresponding components of the unit vector pointing in the same direction. In three dimensions (3 axes of reference: x, y and z) we need three direction cosines: α = angle with respect to x axis, β = angle with respect to y axis and γ = angle with respect to z axis. Referring to figure 7 for a vector OP = a.î + b.ĵ + c.ˆk with magnitude r = (a 2 + b 2 + c 2 ) then the direction cosines are: x,y,z 8
l = cos α = a r m = cos β = b r n = cos γ = c r We can rearrange these to give a = r cos α b = r cos β c = r cos γ and because r 2 = a 2 + b 2 + c 2 we have r 2 = r 2 cos 2 α + r 2 cos 2 β + r 2 cos 2 γ so therefore or cos 2 α + cos 2 β + cos 2 γ = 1 l 2 + m 2 + n 2 = 1 Figure 7: Direction Cosines in Cartesian space. 9