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  Vectors, Physics tutorial

Introduction:

Any physical quantity that needs both magnitude and direction for it to be totally identified is called the vector.

Vector notation:

Generally, vectors are indicated by the letter in bold face type [A, B, C, etc] or by putting the arrow mark or the curly or straight line above letter, or the curly or straight line below the letter, therefore, magnitude of the vector is just denoted by letter without the arrow mark.

Vectors can be graphically represented by directed line segments. Length is chosen, according to some scale, to signify magnitude of the vector, and direction of the directed line segment signifies direction of vector. A vector in plane is a directed line segment. Two vectors are equal if they have same magnitude and direction.

Consider the vector drawn from point A to point B. Point A is known as initial point of vector, and point B is known as terminal point. Symbolic notation for the vector is AB→ (read "vector AB"). Vectors are also symbolized by boldface letters like u, v, and w.

Composition of Vectors:

It is likely to have different vectors representing same physical quantity (like three forces). When these three vector act at same point, the resultant vector can be attained by composition of these different vectors. Vector composition is performed by method of vector addition.

Parallelogram law of vector composition:

619_Parallelogram law of vector composition.jpg

The parallelogram law provides rule for vector addition of vectors A and B. Sum A+B of vectors is attained by placing them head to tail and drawing vector from free tail to the free head.

Let |·| indicate norm of the quantity. Then quantities x and y are said to satisfy the parallelogram law if

|x+y|2 +|x-y|2 + 2|x|2 +2|y|2

If the norm is defined as |f|= √(<f|f>) (the so-called L2-norm), then the law will always hold.

This law is normally stated as:

i) If two vectors acting at the point are denoted by two adjacent sides of the parallelogram drawn from point, then resultant vector will be denoted both in magnitude and direction by diagonal of parallelogram passing through that point.

Addition and subtraction of vectors:

Two or more vectors may be added together to generate their addition. If two vectors have same direction, their resultant has the magnitude equal to sum of their magnitudes and will also have the same direction.

445_Addition of vectors.jpg

Likewise orientated vectors can be subtracted same manner.

1199_subtraction of vectors.jpg

Multiplication of a vector by a scalar:

Scalar-vector multiplication:

Multiplication of the vector by the scalar changes magnitude of the vector, but leaves direction unchanged. Scalar changes size of the vector. Scalar "scales" the vector. For instance, polar form vector...

r = r r^ + θ θ^

multiplied by the scalar a is...

a r = ar r^ + θ θ^

Multiplication of vector by scalar is distributive.

a(A + B) = a A + a B

As a result, rectangular form vector...

r = x i^ + y j^

Multiplied by the scalar a is...

ar = ax i^ + ay j^

Dot product:

Symbol utilized to represent the operation is small dot at middle height (·), that is where name "dot product" comes from. As this product has magnitude only, it is also called as scalar product.

A · B = AB cos θ

Dot product is distributive...

A · (B + C) = A · B + A · C

and commutative...

A · B = B · A

As projection of the vector on to itself leaves its magnitude unchanged, dot product of any vector with itself is square of that vector's magnitude.

A · A = AA cos 0° = A2

Applying this corollary to unit vectors signifies that dot product of any unit vector with itself is one. Additionally, as a vector has no projection perpendicular to itself, dot product of any unit vector with any other is zero.

i^ · i^ = j^ · j^ = k^ · k^ = (1)(1)(cos 0°) = 1

i^ · j^ = j^ · k^ = k^ · i^ = (1)(1)(cos 90°) = 0

By this knowledge we can derive the formula for dot product of any two vectors in rectangular form. Resulting product looks like it is going to be terrible mess, but comprises mostly of terms equal to zero.

A · B = (Ax i^ + Ay j^ + Az k^) · (Bx i^ + By j^ + Bz k^)

A · B = Ax i^  ·  Bx i^  +  Ax i^      ·  By j^  +  Ax i^  ·  Bz k^

 + Ay j^ · Bx i^ + Ay j^ · By j^ + Ay j^ ·Bz k^

 + Az k^ · Bx i^ + Az k^ · By j^ +Az k^ · Bz k^

A · B = AxBx + AyBy + AzBz

The dot product of two vectors is therefore sum of the products of the parallel components. From this we can derive Pythagorean Theorem in three dimensions.

A · A = AA cos 0° = AxAx + AyAy + AzAz

A2 = Ax2 + Ay2 + Az2

The Null Vector:

The null vector is a vector having magnitude equal to zero. It is denoted by O→. The null vector has no direction or it may have any direction. Usually a null vector is either equal to resultant of two equal vectors acting in opposite directions or multiple vectors in different directions.

O→ = A→+(-A→)

Unit Vector:

The unit vector is stated as vector in any specified direction whose magnitude is unity that is 1. A unit vector only specifies direction of the given vector.  Unit vector is signified by any small letter with the symbol of arrow hat (^). The unit vector can be estimated by dividing vector by its magnitude. For instance unit vector of the vector A is given by:

Û= A→/A

In three dimensional coordinate system unit vectors i^, j^, k^ having direction of positive X-axis, Y-axis and Z-axis are utilized as unit vectors. These unit vectors are mutually perpendicular to each other.

Components of a Vector:

Components of a vector in terms of unit vectors:

The vector is defined by its magnitude, r and its direction, OP→. It could also be defined by its two components in the OX and OY directions. What we are saying here is that is vector acting along a plane and could be resolved into its components.

Thus: b→P→

is equivalent to a vector in the OX direction plus a vector in the OY direction. i.e. OP→ (along OX axis) (along OY axis)

If we take to be unit vector in the OX direction then

a→ =  ai^

Likewise, if we define j to be a unit vector in the OY direction, then

b→= bj^

OP→

r→ =a i^+ bj^

where î and j^ are the unit vectors in the 0X and 0Y directions respectively. Sign (called cappa) signifies the unit vector.

309_Components of vectors.jpg

Component of a Vector in Terms of Polar Coordinates:

In Polar coordinates the vector as shown in Figure resolved along the OX and OY axes thus:

p→ From the end point of vector draw the perpendicular PC and PD on X and Y-axes respectively. Then, OC and OD represent resolved parts of vector in magnitude and direction. Therefore we have

OC = OP Cos 2 = r Cos 2

OD = OP Sin 2 = r Sin 2

and OC2 + OD2 = OP2= r2 (Cos22 + Sin22) = r2

Now, OP→= rcos θ and OD→ rsin θ are the components of vector in polar coordinates.

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