Vectors

definition

Scalar Quantities

A scalar quantity is a quantity which is defined by only magnitude. Some examples of scalar quantities are Mass, Charge, Pressure, etc.

definition

Vector quantities

Vector quantities are those which have both magnitude and direction and obey vector laws of addition. Some examples of vectors are displacement, velocity, force, etc.
A quantity is called a vector only if it follows all the above three conditions. For example, current is not a vector despite having both magnitude and direction because it does not follow vector laws of addition.

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Scalars and Vectors

Following are some differences listed between scalars and vectors.

S.No.	Scalars	Vectors
1.	Have only magnitude	Have both magnitude and direction
2.	Algebra: Same as real numbers	Algebra: Follow vector laws of addition
3.	Examples: Mass, charge, etc	Examples: velocity, force, electric field. etc.

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Free and fixed vectors

Free vector:
A free vector is a vector whose action is not confined to or associated with a unique line in space. Examples: velocity vector, electric field vector, etc.
Sliding vector:
A vector that can be applied at any point on a body as long as it is along its original line of action and doesn't change its effect in the body as a whole. Example: force vector, etc.
Fixed vector:
Fixed vector is that vector whose initial point or tail is fixed. Example: position vector, etc.

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Calculating Unit vector

A unit vector is a vector whose magnitude is 1 represented by a lowercase letter with a hat.
For example,

\hat{i}

is a unit vector.

\begin{matrix} 1 \\ \sqrt{3} \end{matrix} \hat{i} + \begin{matrix} 1 \\ \sqrt{3} \end{matrix} \hat{j} + \begin{matrix} 1 \\ \sqrt{3} \end{matrix} \hat{k}

is also a unit vector since its magnitude is

\sqrt{{(\frac{1}{\sqrt{3}})}^{2} + {(\frac{1}{\sqrt{3}})}^{2} + {(\frac{1}{\sqrt{3}})}^{2}}

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Zero vector

A zero or null vector is a vector whose magnitude is 0 and has unspecified arbitrary direction. For example when a body projected vertically from the ground returns to the ground, its vertical displacement is a zero or null vector.

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Equality of vectors

Two vectors are said to be equal if (i) they have the same magnitude and (ii) are in the same direction. If we shift B parallel to A then it will completely superimpose A i.e it has same length and are in the same direction as A, so A=B

definition

Rectangular coordinate system

Rectangular coordinate system is a coordinate system that specifies each point uniquely in a plane by a triplet of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The coordinates of a point are often represented as

x, y

and

z

. The point of meeting of the three axis is known as origin. A vector is represented in rectangular coordinate system as:

\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

where

A_{x}, A_{y}

and

A_{z}

are distances of the vector measured along the

x, y

and

z

axis respectively,

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Triangle Law of Vector addition

Triangle law of vector addition states that when two vectors are represented by two sides of a triangle in magnitude and direction taken in same order then third side of that triangle represents in magnitude and direction the resultant of the two vectors.

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Associative, distributive and commutative law of vector addition

Vector addition follows:
1. Associative law:

\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}

2. Commutative law:

\vec{A} + \vec{B} = \vec{B} + \vec{A}

3. Distributive law:

k (\vec{A} + \vec{B}) = k \vec{A} + k \vec{B}

(

k

is a scalar)
Note:
There are two other forms of distributive law-

\vec{A} . (\vec{B} + \vec{C}) = \vec{A} . \vec{B} + \vec{A} . \vec{C}

\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}

definition

Parallelogram Law of Vector Addition

Parallelogram Law of Vector Addition states that when two vectors are represented by two adjacent sides of a parallelogram by direction and magnitude then the resultant of these vectors is represented in magnitude and direction by the diagonal of the parallelogram starting from the same point.

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Law of sines and law of cosines

Law of cosines is used to find the magnitude sum of two vectors being added.

\vec{A} + \vec{B} = \vec{C}

| \vec{C} | = \sqrt{| \vec{A} |^{2} + | \vec{B} |^{2} + 2 | \vec{A} | | \vec{B} | \cos (θ_{C})}

where

θ

is the angle between

\vec{A}

and

\vec{B}

.

Law of sines is used to find the relation between magnitudes of vectors and angles between them.

\frac{| \vec{A} |}{\sin θ_{A}} = \frac{| \vec{B} |}{\sin θ_{B}} = \frac{| \vec{C} |}{\sin θ_{C}}

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Polygon Law of Vector Addition

Polygon law of vector addition states that if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

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Addition of Vectros in Rectangular Co-ordinate System

Let's take two vectors in rectangular co-ordinate system:

\vec{a} = a_{1} \hat{i} + a_{2} \hat{j}

\vec{b} = b_{1} \hat{i} + b_{2} \hat{j}

Addition of Vectors:

\vec{a} + \vec{b} = (a_{1} + b_{1}) \hat{i} + (a_{2} + b_{2}) \hat{j}

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Magnitude of vector in rectangular coordinate system

Let a vector be defined as

\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}

Then, its magnitude is given by:

| \vec{A} | = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}

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Dot Product

The dot product of two vectors

\vec{a}

and

\vec{b}

is defined as:

\vec{a} . \vec{b} = a b c o s θ

example

Dot Product of Parallel and Perpendicular Vectors

\vec{a} . \vec{b} = 0

(For Perpendicular Vectors)

\vec{a} . \vec{b} = a b

(For Parallel Vectors)

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Component of a Vector

Since

\vec{a} . \vec{b} = a b c o s θ

Projection of vector

a

b

is:

a c o s θ = \vec{a} . \hat{b}

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Product of a scalar and a vector

When a vector is multiplied by a scalar, its magnitude is multiplied by the direction of the scalar.
If a vector is defined as

\vec{A} = | A | ∠ A

where

∠ A

is defined as the angle with respect to a given fixed line.
Then,

k \vec{A} = k | A | ∠ (A \pm 180^{o})

k < 0

k \vec{A} = k | A | ∠ (A)

k > 0

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Dot Product of Two Vectors expressed in rectangular coordinate system

Representation of vectors in rectangular coordination system:

\vec{a} = a_{1} \hat{i} + a_{2} \hat{j}

\vec{b} = b_{1} \hat{i} + b_{2} \hat{j}

Their dot product is:

\vec{a} . \vec{b} = a_{1} b_{1} + a_{2} b_{2}

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Angle Between Vectors

c o s θ = \frac{\vec{a} . \vec{b}}{| a | | b |}

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Linear combination of orthogonal vectors

Let us define a set

S

of three perpendicular vectors that can be defined in the rectangular coordinate system.

S = {{\vec{S}}_{1}, {\vec{S}}_{2}, {\vec{S}}_{3}}

Then, any vector

\vec{A}

in the rectangular coordinate system can be represented as a linear combination of the three vectors in the set S. This is done by taking a dot product along each component.
Example:
A vector is defined as

\vec{A} = 3 \hat{i}

. Rewrite

\vec{A}

as a linear combination of

{\vec{S}}_{1} = 2 \hat{i} + \hat{j}

and

S_{2} = \hat{i} - 2 \hat{j}

.
Unit vectors are:

\hat{S_{1}} = \frac{2 \hat{i} + \hat{j}}{\sqrt{5}}

\hat{S_{2}} = \frac{\hat{i} - 2 \hat{j}}{\sqrt{5}}

Component of A along

S_{1}

is :

A_{1} = \vec{A} . \hat{S_{1}} = 6 / \sqrt{5}

Component of A along

S_{2}

is :

A_{2} = \vec{A} . \hat{S_{2}} = 3 / \sqrt{5}

Then,

\vec{A} = A_{1} \hat{S_{1}} + A_{2} {\hat{S}}_{2}

Solving,

\vec{A} = \frac{6}{5} {\vec{S}}_{1} + \frac{3}{5} {\vec{S}}_{2}

Note:
1. Orthogonality of the given vectors is a necessary condition to write a vector as a linear combination along the vectors.
2. Number of orthogonal unit vectors is equal to the dimension of the coordinate system.

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Cross Product of Vectors

The Cross Product

a \times b

of two vectors is another vector that is at right angles to both.

\vec{a} \times \vec{b} = a b s i n (θ) \hat{n}

Where

\hat{n}

is perpendicular to both the vectors.

definition

Right hand thumb rule

Direction of the resultant of the cross product of two vectors can be found using the right hand thumb rule. It is shown in the attached figure.
For example, if

\vec{a}

is in east direction and

\vec{b}

is in north direction, then

\vec{a} \times \vec{b}

is in the upward direction.

formula

Cross Product of Vectors

\vec{a} \times \vec{b} = 0

(For parallel vectors)

\vec{a} \times \vec{b} = a b \hat{n}

(For perpendicular vectors)
(Where

\hat{n}

is perpendicular to both the vectors)

definition

Cross Product of Unit Vectors

\vec{a} \times \vec{b} = \hat{n}

(For perpendicular vectors)
(where

\hat{n}

is a unit vector perpendicular to both the vectors)

example

Cross Product of Vectors

Representation of vectors in rectangular coordination system:

\vec{a} = a_{1} \hat{i} + a_{2} \hat{j}

\vec{b} = b_{1} \hat{i} + b_{2} \hat{j}

Their cross product is:

\vec{a} \times \vec{b} = (a_{1} b_{2} - a_{2} b_{1}) \hat{n}

example

Angle Between Vectors using cross-product

Angle between vectors can be determined using cross-product by:

s i n (θ) = \frac{| \vec{a} \times \vec{b} |}{| a | | b |}

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Area of parallelogram using cross product

Area of parallelogram is given by:

shortcut

Right Handed System of Vectors

When two vectors

u

and

v

are kept tail-to-tail,then placing the right hand along the direction of

u

, and curling the fingers in the direction of the angle

v

makes with

u

, the thumb points in the direction of

u \times v

.
A three-dimensional coordinate system in which the axes satisfy the right-hand rule is called a right-handed coordinate system, while one that does not is called a left-handed coordinate system.

TB - UG

Monday, July 31, 2017

Vectors

Scalar Quantities

Vector quantities

Scalars and Vectors

Free and fixed vectors

Calculating Unit vector

Zero vector

Equality of vectors

Rectangular coordinate system

Triangle Law of Vector addition

Associative, distributive and commutative law of vector addition

Parallelogram Law of Vector Addition

Law of sines and law of cosines

Polygon Law of Vector Addition

Addition of Vectros in Rectangular Co-ordinate System

Magnitude of vector in rectangular coordinate system

Dot Product

Dot Product of Parallel and Perpendicular Vectors

Component of a Vector

Product of a scalar and a vector

Dot Product of Two Vectors expressed in rectangular coordinate system

Angle Between Vectors

Linear combination of orthogonal vectors

Cross Product of Vectors

Right hand thumb rule

Cross Product of Vectors

Cross Product of Unit Vectors

Cross Product of Vectors

Angle Between Vectors using cross-product

Area of parallelogram using cross product

Right Handed System of Vectors

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