Quantities That Have Direction As Well As Magnitude Are Called As Vectors. Examples Of

Vectors

Quantities that have direction as well as magnitude are called as vectors. Examples of vectors are velocity, acceleration, force, momentum etc.

Vectors can be added and subtracted. Let a and bbe two vectors. To get the sum of the two vectors, place the tail of bonto the head of aand the distance between the tail of a and b is a+b.

Multiplication of a vector by a positive scalar k multiplies the magnitude but leaves the direction unchanged. If k=2 then the magnitude of adoubles but the direction remains the same.

Dot product of two vectors is the product of a vector to the projection of the other vector on the vector. a. bis called the dot product of the two vectors.

a. b = . If the two vectors are parallel, then a. b =And if the two vectors are perpendicular to each other,

then a. b = 0

Cross Product of any two vectors is defined by ab= c =, where is a unit vector (vector of length 1) pointing perpendicular to the plane of a and b. But as there are two directions perpendicular to any plane, the ambiguity is resolved by the right hand rule: let your fingers point in the direction of the first vector and curl around (via the smaller angle) towards the second; then your thumb indicates the direction of.

Unit vector is a vector whose magnitude is 1 and point is a particular direction. Without loss of generality, we can assume to be three distinct unit vectors along the x, y, and z-axis relatively.

Then,

and

Also,

In cylindrical coordinate systems, a vector, where are the unit vectors of the coordinate system.

In spherical coordinate system, a vector, where are the unit vectors of the coordinate system.

Vector Practice

1. Consider three vectors:

a. Draw the three vectors.

b. What is the length or magnitude of, and?

Length of the vector == 3

Length of the vector == 4

Length of the vector ==10

c. What is the angle between and, and, and?

We know that =,

So, So,

Now,=

=

=

So, either or .

But notice that the y-component of is positive and the x-component negative. So it is in the 2nd quadrant and the vector is in the first quadrant. So, the angle between them cannot be more than. So the angle between and is .

Similarly, the angle betweenandis

,,

either or . But , as none of the components of and are negative. So,

Similarly, the angle between andis

,,

either or . But , as none of the components of and are negative. So,

2. Consider three vectors:

a. What is the length or magnitude of, also written as?

Length of the vector ===

b. Write the expression for 2.

2= 2() = 2() =

c.What is?

= =

=

=

d. What is ?

= =

=

=

e. What is ?

=

==

=

=

=

f. What is the magnitude of ?

Magnitude of =

Let us name =

Then magnitude of =

===

g. What is?

=

=

=

=16

Note:

is the dot product of the two vectors.

=, where is the angle formed by and when they are place tail-to-tail.

h. What is the angle between and ?

We know that = ,

So,

So,

So

i. Does equal?

Yes. We can see that is a scalar (hence the alternative name scalar product). So the dot product is commutative. That is =

j. How is and related?

Using the formula for the cross product,

=

=

= ---1

And =

=

= ---2

Equating 1) and 2), we can clearly see that

() = -()

k. Give an example of the use of dot product in Physics and explain.

Mechanical work is a dot product between the force and the displacement.

It means that the work is amount of displacement times the projection of force along the displacement.

l. Give an example of the use of cross product in Physics and explain.

Magnetic force is a good example of a cross product of velocity of a charged particle times the magnetic field.

It means that the force is perpendicular to both and, and that the magnitude of the force is equal to the area of the parallelogram they span.

m. Imagine that the vector is a force whose units are given in Newtons. Imagine vector is a radius vector through which the force acts in meters. What is the value of the torque, in this case?

Torque is the cross product between the radius vector and the force vector, .

So, in this case

=

=

=

=

=

=

n. Now imagine that continues to be a force vector and is a displacement vector whose units are meters. What is the work done in applying force through a displacement?

Work done in applying force through a displacementis

=

=

=

=8 joules

o. What is the vector sum of a vector given by 40 m, 30 degrees and a vector given by 12 m, 225 degrees? Use the method of resolving vectors into their components and then adding the components.

Let us start by resolving the vectors into their x-components and y-components.

It can be assumed that the angles given in the question are w.r.t the x-axis.

For:

The magnitude of = is 40 and the angle is 30o

So, as the projection of along the x-axis = cos

The projection of along the y-axis = sin

Similarly for:

So, += =

=

3. Consider three vectors:

a. Find.

First we shall find

= =

=

=

Let us name

Now, =

=

=

= 3

b. Find.

First we shall find

=

=

=

=

=

Let us name

Now, =

=

=

= 52

c. Find.

First we shall find

= =

=

=

Let us name =

Now =

=

=

=

=

=