Introducing vector

A vector can be defined as a quantity that has magnitude and direction. The point A from where the vector begins is known as its initial point, and the point B where it ends is known as its terminal point. The distance between initial and terminal points of a vector is known as the magnitude (or length) of the vector, represent as | |, or ||, or a. The arrow shows the direction of the vector.

Position vector:

Consider a point P in space, which has coordinates (x, y, z) with respect to the origin O (0, 0, 0). Then, the vector that has O and P as its initial and terminal points, correspondingly, is known as the position vector of the point P with regard to O.

Direction cosines:

Consider the position vector (or ) of a point P(x, y, z) . The angles made by the vector with the positive directions of x, y and z-axes correspondingly, are known as its direction angles. The cosine values of these angles, i.e., cos, cos and cos are known as direction cosines of the vector and normally represented by l, m and n, in that order.

Types of Vectors:

Zero Vector

A vector whose initial and terminal points coincide, is referred to as a zero vector (or null vector) . Zero vector cannot be designated a definite direction as it has zero magnitude. Or, on the other hand otherwise, it may be regarded as having any direction. The vectors stand for the zero vector,

Unit Vector

A unit vector is the vector whose magnitude is unity (i.e., 1 unit) is known as a unit vector.

Coinitial Vectors

Two or more vectors that has the same initial point are known as coinitial vectors.

Collinear Vectors

Two or more vectors are said to be collinear if they are parallel to the same line, regardless of their magnitudes and directions.

Equal Vectors

Two vectors and are said to be equal, if they have the same magnitude and direction irrespective of the positions of their initial points.

Negative of a Vector

A vector whose magnitude is the same as that of a given vector for an example but whose direction is opposite to that of it, is known as negative of the given vector.

Scalar (or dot) product of two vectors :

a . b = | a || b | cosθ, where, θ is the angle between a and b , 0 ≤θ≤π

Observations with regards to Scalar Products

1. a . b is a real number.

2. Let a and b be two nonzero vectors, then a . b = 0 if and only if a and b are perpendicular to each other. i.e. 

a . b = 0⇔ a ⊥ b 

3. If θ= 0, then a . b = | a || b |

In particular, a . a = | a |2, as θ in this case is 0.

4. If θ= π, then a . b = - | a || b |

In particular, a .(- a ) = - | a |2, as θ in this case is π.

5. With regards to view of the Observations 2 and 3, for mutually perpendicular unit vectors i, j and k, we have

i.i= j.j= k.k=1,

i.j= j.k= k.i=0

6. The angle between two non zero vectors a and b is given by

Cos θ = a . b | a || b | , θ = cos-1( a . b | a || b |)

7. The scalar product is commutative. i.e.

a . b = b . a

Multiplication of a Vector by a Scalar

Let a be a given vector and λ a scalar. Then the product of the vector a by the scalar λ, represented as λ a is known as the multiplication of vector a by the scalar λ. Observe that, λ a is as well a vector, collinear to the vector a . The vector λ a has the same direction (or opposite) to that of vector a with regards to the way the value of λ is positive (or negative). Again, the magnitude of vector λ a is | λ | times the magnitude of the vector a , i.e.,

| λ a | = | λ | | a |

Vector product :

The vector product of two nonzero vectors a and b , is represented by a x b and defined as

a x b = | a || b | sin θ n ,

where, θ is the angle between a and b , 0 ≤ θ ≤ π and n is a unit vector perpendicular to both a and b , such that a , b and n form a right handed system. i.e., the right handed system rotated from a to b moves in the direction of n.

Scalar triple product

Let a , b and c be three vectors. Then the scalar ( a x b ). c is known as scalar triple product of a , b and c and is represent ed by [ a b c ]

∴ [ a b c ] = ( a x b ). c 

If a , b and c stand for the three co-terminus edges of a parallelepiped then its volume = [ a b c ]

Properties of Scalar Triple Products

If a , b and c are cyclically permuted, the value of the scalar triple product remains the same.

[ a b c ] = [ b c a ] = [ c a b ]

The position of dot and cross can be interchanged, in so far as the cyclic order of vectors remains the same

a .( b x c ) = c .( a x b ) = ( a x b ). c

The value of scalar triple product remains the same in magnitude, but alters the sign, if the cyclic order of a , b and c is altered.

The scalar triple product of three vectors is zero if any two of the given vectors are equal.

For any three vectors a , b and c and a scalar λ, we have

[λ a b c ] = λ[ a b c ] The scalar triple product of three vectors is zero if any two of the given vectors are parallel or collinear.

Vector triple product :

Let a , b and c be three vectors. Then the vector ( a x b )x c is known as vector triple product of a , b and c : ( a x b )x c = ( a . c ) b - ( b . c ) a 

( a x b )x c is coplanar with a and b and is perpendicular to c .

Again, the vector triple products are not associative.

Cartesian Components of a Vector

Consider a Cartesian coordinate system made up of an origin, and three mutually perpendicular coordinate axes. Such a system is said to be right-handed if, when looking along the direction, a 90o clockwise rotation about Oz is needed to take Ox into Oy . If not, it is said to be left-handed. In physics, it is a rule to constantly make use of right-handed coordinate systems.

It is convenient to define unit vectors, ex ,ey , and ez , parallel to Ox ,Oy , and Oz , correspondingly. By the way, a unit vector is a vector whose magnitude is unity. The position vector, r , of a few general point p whose Cartesian coordinates are ( x, y,z ) is then given by

r = Xex + yey + zez

On the other hand, we can obtain from O to P by moving a distance x parallel to Ox , then a distance y parallel to Oy , and then a distance z parallel to Oz . In the same way, if a is an arbitrary vector then

a = axex + ayey + azez,

where ax,ay , and az are referred to as the Cartesian components of a . It is conventional to write . It follows that , , and . Of course, .

According to the three-dimensional generalization of the Pythagoras theorem, the distance is given by

By analogy, the magnitude of a general vector a takes the form

If andthen it is readily illustrated that

In addition, if is a scalar then it is clear that

Rotational Invariance

One significant feature of the dot product which we didn't mention in the previous section is its invariance under rotations. In other words, if we take a pair of vectors in the plane and rotate them both by the same angle (assuming, for instance, that the vectors are sitting on a record, and rotate the record), their dot product will remain the same. Consider the length of a single vector (which is given by the dot product): if the vector gets rotated about the origin by a few angle, its length will not alter--even though its direction can alter quite dramatically. In the same way, from the geometric formula for the dot product, we observe that the result depends only on the lengths of the two vectors and the angle between them. None of these quantities alters when we rotate the two vectors together, therefore neither can their dot product. This is what we mean when we say that the dot product is invariant under rotations.

Vector subtraction

Before we define subtraction, we define the vector −a, which is the opposite of a. The vector −a is the vector with an equivalent magnitude as a but that is pointed in the opposite direction.

We define subtraction as addition with the opposite of a vector:

b−a=b+(−a).

This is similar to turning vector a around in the application of the above rules for addition. Can you observe how the vector x in the below figure is equal to b−a? Observe the way this is the same as stating that a+x=b, just like with subtraction of scalar numbers.

Scalar multiplication

Given a vector a and a real number (scalar) λ, we can form the vector λa as follows. If λ is positive, then λa is the vector whose direction is the same as the direction of a and whose length is λ times the length of a. In this case, multiplication by λ simply stretches (if λ>1) or compresses (if 0<λ<1) the vector a.

If, on the contrary, λ is negative, then we have to take the opposite of a before stretching or compressing it. In other words, the vector λa points in the opposite direction of a, and the length of λa is |λ| times the length of a. No matter the sign of λ, we notice that the magnitude of λa is |λ| times the magnitude of a: ∥λa∥=|λ|∥a∥.

Scalar multiplications satisfies a lot of of the same properties as the normal multiplication.

1. s(a+b)=sa+sb (distributive law, form 1)

2. (s+t)a=sa+ta (distributive law, form 2)

3. 1a=a

4. (−1)a=−a

5. 0a=0

In the last formula, the zero on the left is the number 0, whereas the zero on the right is the vector 0, which is the specific vector whose length is zero.

If a=λb for some scalar λ, then we say that the vectors a and b are parallel. If λ is negative, A few people say that a and b are anti-parallel, but we will not make use of that language.

We were able to describe vectors, vector addition, vector subtraction, and scalar multiplication without reference to any coordinate system. The usefulness of such purely geometric reasoning is that our results hold generally, independent of any coordinate system in which the vectors are found. However, sometimes it is useful to express vectors in terms of coordinates, as explained in a page about vectors in the standard Cartesian coordinate systems in the plane and in three-dimensional space. View more on psalmfresh.blogspot.com for more educating and tricks article Thank you