Mathematical interpretation of everyday reality of all linear algebra boils down to matrices and all of matrices boils down to vectors – Anonymous

Vectors, what really are they? To answer this question, there are three main ways of looking at it. One is the physics perspective; one is the Computer Science perspective and one is the Mathematical perspective. As always, we’re going to explore this topic both on an intuitive and mathematical level. Let’s tackle the Physics perspective in this article which is arguably the most trivial and easy to understand. More complex vector equations and operations will however not be included in this article (although I personally love linear algebra and higher applications of vector fields and tensor mechanics).

A vector is a physical quantity that has both magnitude and direction. Some examples of vectors are displacement, velocity, force, electric field, magnetic field, momentum, acceleration etc. Usually, in physics a vector is represented by a capital letter with an arrow over it. The magnitude of vector A is represented by |A|. A vector can be represented by an arrow parallel to the direction of the vector. The length and direction of the arrow thus represent the magnitude and direction of the vector respectively. To find the angle between two vectors, both vectors are drawn from one point in such a manner that tails of both the vectors coincide.

There are numerous kinds of vectors which we’ll encounter. A vector of unit magnitude is called a unit vector and the notation for it in the direction of A is “A cap” or “A caret”. Thus, A vector=|A|(A cap).Unit vectors along the x, y and z axes are called basis vectors for the standard three-dimensional coordinate system are represented as i cap, j cap and k cap respectively.

A vector having zero magnitude is called a null vector or a zero vector. The zero vector has no specific direction, the position of the vector of origin is a zero vector and they are only of mathematical importance having no physical meaning. Vectors are equal if they have the same magnitude and direction. The negative of any vector is a vector having equal magnitude acting in the opposite direction.

Vectors having the same initial points are called concurrent or co-initial vectors. Vectors that lie in the same plane are known as co-planar vectors.Vectors are said to be orthogonal if the angle between them is 90°. Hence, the unit vectors i cap, j cap and k cap are also called orthogonal unit vectors.

Scalars like distance, speed, density, energy etc can be added, subtracted, multiplied or divided using basic mathematical rules. However, when we come to vectors the things get slightly more interesting. The product of a vector A and a scalar n is nA whose magnitude is n times the magnitude of A and is in the same direction of A if n is positive and is in the opposite direction if n is negative. Further, if m and n are two scalars, (m+n)A= mA+nA and m(nA)=n(mA)=(mn)A. The division of a vector A by a non-zero scalar n is nothing but the multiplication of A by 1/n. Addition and subtraction of vectors is where things start to get slightly more mathematical. But still, as I promised its very trivial stuff.

There are a few ways to compute the addition or subtraction of vectors, the most common one is the parallelogram law of vector addition. Let R be the resultant of two vectors A and B. According to this law, the resultant R is the diagonal of the parallelogram of which A and B are the adjacent sides. Magnitude of R is given by, R=(A^{2}+B^{2}+2ABcosφ)^{1/2}. Here φ is the angle between A and B. The angle of R can be found out by angle α or β of R with A or B. Tanα=Bsinφ/A+Bcosφ and tanβ=Asinφ/B+Acosφ. It can be mathematically proven using range of trigonometric functions that R is maximum when φ=0° and minimum when φ=180°. R_{max}=A+B and R_{min}=A-B.

Another way of computing this is the triangle law of vector addition. According to this law, if tail of one vector is placed at the head of the other, their sum or resultant R is drawn from the tail end of the first to the head end of the other. Therefore, R=A+B=B+A.

Subtraction of vectors is nothing but reversing the direction of one vector and then adding it. Hence, A-B=A+(-B). In this case, the angle between the two vectors is 180-φ and hence the resultant A-B=(A^{2}+B^{2}-2ABcosφ)^{1/2}. For the direction of this resultant vector, we can find tanα=Bsinφ/A-Bcosφ and tanβ=Asinφ/B-Acosφ.

To classify as a vector, a physical quantity must not only have direction and magnitude, but it should also follow the triangle/parallelogram law of vector addition. For example, current has magnitude and direction, but it is still classified as a scalar as it doesn’t follow the triangle/parallelogram law of vector addition and is just added up like any other scalar. This concept can be generalised in the polygon law of vector addition which states that if a vector polygon be drawn placing the tail end of each succeeding vector at the head or the arrow end of the preceding one, their resultant R is drawn from the tail end of the first to the head or arrow end of the last. Thus, here R=A+B+C+D.

Two or more vectors which when compounded in accordance with the parallelogram law of vector R are said to be the components of vector R. Components of a vector can be made in any mutually perpendicular directions. But in general, we mostly use the standard coordinate system. Thus, the vector R can also be written as R = R_{x}(i cap)+ R_{y}(j cap)+ R_{z}(k cap). Here, R_{x}, R_{y, }R_{z}are components of R along the x, y and z axes respectively. The magnitude of R can be given as R=(R_{x}^{2}+R_{y}^{2}+R_{z}^{2})^{1/2}.

The vector R makes an angle α with the x axis, β with the y axis and γ with the z axis. Here, cosα=R_{x}/R, cosβ=R_{y}/R and cosγ=R_{z}/R. Cosα, cosβ and cosγ are also called direction cosines of R with the x, y and z axes respectively. An important identity to be noted here is that cos^{2}α+cos^{2}β+cos^{2}γ=1.

Another interesting concept that stems up is the position vector. To locate the position of any point P in a plane or space, generally a fixed point of reference called the origin O is taken. The vector OP is the position vector of P with respect to O. If the coordinates of P are (x, y), then the position vector of point P with respect to O is OP=r=x(i cap) + y(j cap). However, there are two key points to be noted over here. For a point P, there is one and only one position vector with respect to O and the position vector of point P changes if the position of point O is changed.

Coming on to the multiplication of vectors, it is here that things start to get more interesting especially for Physics and Math lovers. The product of two vectors, is of two kinds i.e. the dot product and the cross product. The dot product, also called the scalar product of two vectors A and B is denoted by A.B and is read as A dot B. It is defined as the product of the magnitudes of the two vectors A and B and the cosine of their included angle φ. Thus, A. B=|A||B|cosφ.

Few properties of the dot product are to be kept in mind. They are commutative and distributive. Hence, A.B=B.A and A.(B+C)=A.B+A.C; A.A=A^{2}; i.i=j.j=k.k=1; i.j=j.k=k.i=0. Component of B along A is A.B/|A| and component of A along B is A.B/|B|. Two vectors are perpendicular if their dot products are zero. Another important computational identity for the dot product is (a_{1}(i cap)+b_{1}(j cap)+c_{1}(k cap)).(a_{2}(i cap)+b_{2}(j cap)+c_{2}(k cap))=a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2.}The dot product is also called as the scalar product as the result of a dot product is always a scalar. There are many uses of the dot product in physics such as: W=F.S, P=F.V, U_{E}=M.B, U_{B}=M.B and many more.

Coming onto the cross product, it is denoted by AxB and is read as A cross B. It is defined as a third vector C which is perpendicular to both A and B and whose magnitude is equal to the product of the magnitudes of the vectors A and B and the sine of their included angle φ. Thus, C=AxB=|A||B|sinφ(n cap). Here (n cap) is the unit vector in the direction of vector C.

There are a few important properties of the cross product. AxB = -BxA, the cross product of two parallel or anti parallel vectors in zero. Thus ixi=jxj=kxk=0. The cross product is also distributive. Hence, Ax(B+C) = AxB + AxC. Another important identity is: ixj=k, jxk=i, kxi=j. We can compute the cross product with the help of a determinant as well.

There are many applications of the cross products in physics itself such as: τ=rxF, L=rxP, F_{B}=q(vxB) and many more. Another interesting result is that two vectors can be proved as parallel or anti parallel to one another if both the vectors bear a constant ratio. For instance, vector A=a_{1}(i cap)+b_{1}(j cap)+c_{1}(k cap) is parallel to vector B= a_{2}(i cap)+b_{2}(j cap)+c_{2}(k cap) if a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}= constant value. If this constant value is positive, the vectors A and B are parallel and if the constant is negative, the vectors A and B are anti parallel.

A physical quantity need not always be a vector or a scalar. We can generalize it as a tensor. A tensor is defined as a mathematical object that describes a multilinear relationship between sets of algebraic objects related to a vector space. A tensor is characterised by its rank, a scalar is a tensor of rank 0 while a vector is a tensor of rank 1.

** **Tensors are used in many fields of physics such as stress, elasticity, fluid mechanics, moment of inertia, electromagnetism, magnetic susceptibility, premittivity, general relativity and many more. ** **In physics, a tensor can occur at multiple points of an object and the tensors may vary from one location to another. This paved the way for the concept of a tensor field. A tensor field assigns a tensor to each point in space. Several times tensor fields are just simply called tensors. Tensor fields are used in many fields of physics and maths such as Differential geometry, general relativity, algebraic geometry and many more.

Moment of inertia has two forms, a scalar form (when the axis of rotation is known) and a more general tensor form which does not require the axis of rotation. Stress is neither a scalar nor a vector, it is a tensor. Area can behave either as a scalar or a vector depending upon the circumstances. Vectors associated with linear or directional effect are called polar vectors and those associated with rotation about an axis are called axial vectors. For example; force, linear velocity, linear acceleration etc are polar vectors whereas angular velocity, angular acceleration etc are axial vectors. Another couple of interesting proofs can be derived. The area of a triangle can be computed as |AxB|/2, similarly the area of a parallelogram would be |AxB|.

Another interesting application of the multiplication of vectors is the scalar triple product. Here, A. (BxC)=B.(CxA)=C.(AxB) is called the scalar triple product of the vectors A, B and C. It is a scalar quantity and the volume of a parallelepiped bounded by the three vectors is given by the scalar triple product of these vectors. A parallelepiped is nothing but a solid body who’s each face is a parallelogram. The scalar triple product of A, B and C can also be written as [ABC]. If the scalar triple product of three vectors is zero, this implies that they are coplanar.

Another interesting concept I’d like to mention is of the vector triple product. The vector triple product is interpreted as the cross product of one vector with the cross product of the other two.** **The following relationship is called the triple product expansion: Ax(BxC) = (A.C)B-(A.B)C.

Vectors are frequently used in physics, maths and engineering fields but there are also quite a few day to day applications of vectors. For example in sports when a player throws a ball or hits it, vectors are inevitably involved in that situation. In such a case, we can determine the range, maximum height and time period of flight of the ball. Even while crossing a river, a boat often drifts due to the flow of water. Here too vectors are involved.

Vectors are used in many sports and also in much cooler applications such as the determination of force required to move a body resting on a surface with friction. Vectors also have many applications in the fields of thermodynamics and electromagnetism. There are many such more uses of vectors in our everyday life and we have been knowingly or unknowingly using vectors all the time! That was the Physicists perspective and I’ll soon be back with the more generalised mathematical form of it.

Very good article Pranav, keep it up !

Thank you very much !

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