2. resolution of vectors

Relative velocity

Relative velocity is the velocity of one object with respect to another object.

Every motion is relative as it has to be observed with respect to an observer. Relative velocity is a measurement of the velocity of an object with respect to another observer. He is defined as the time rate of change of relative position of one object with respect to another.

Most of you must have traveled on a train someday, you would have probably noticed that the other train crossing your train from the opposite side seems to be moving at a very high speed, It is because the relative velocity of the other train is the sum of your train speed and its actual speed.

Unit vectors

Unit vector is a vector with unit magnitude. It is used to specify direction.  i, j and k are the unit vectors along x, y and z-direction respectively.

• We can define the unit vector of any vector. Unit vector of a vector is of unit length and along with the same direction as that of the vector

• If we have a vector A= A_{x i + Ay j+ Az k}

And the magnitude of the vector A is given by |A|= √(Ax^2+Ay^2+Az^2 )

Then unit vector of A=  A/(|A|)

Below is one the example

Resolution of a vector in a plane

Consider a vector in x-y plane  a^→= ax i+ ay j

Here ax and ay is the x and y component of Vector A.

Suppose vector A makes an angle θ with the horizontal then

ax= |a| cosθ    and  ay= |a| sinθ  where |a|= magnitude of vector a

We have two ways to specify a vector

1.  When we have the magnitude of the vector A and angle θ made with the horizontal.

A= Acosθ i+ Asinθ j

2. When we have its horizontal and vertical components Ax and Ay.

A= Ax i + Ay j

If there are two ways to represent the same vector, there must be some relation connecting the two representation

The magnitude of vector A=√(Ax^2+Ay^2 )

tanθ= Ay/Ax    ;  θ= tan^(-1) (Ay/Ax)

Similarly, we can resolve any vector quantity in its horizontal and vector component as shown above.

In the figure below the velocity vector with magnitude 50m/s which is at 60 degrees from the horizontal is resolved.

Another example of the resolution of force. Suppose a dog is held by its neckband with a force of 60 N acting at an angle of 40 degrees with the horizontal.

Vector Addition by rectangular components.

Finding the resultant of the vectors by using the component method.

Suppose we have two vectors given below and we need to find the resultant of these two vectors.

• First we will resolve the two vectors in their components and write them in vector notation.
• Vector A makes an angle of 70 with the positive horizontal direction( +x axis) and has a magnitude A= 3.6 m.

Ax= A cos 70  = 3.6 cos70= 1.23

Ay= A sin 70 = 3.6 sin 70= 3.38

        So vector A= 1.23 i + 3.38 j

• Similarly, vector B has a magnitude of 2.4 and makes an angle of 30 with the negative x-axis. When we resolve vector B, the components Bx and By are along the -x and -y-axis as this vector is in the third quadrant.

Bx= B cos 30= 2.4 cos 30=2.078

By= B sin 30=2.4 sin 30= 1.2

vector B= 2.078 (-i) + 1.2 (-j)= -2.078 i -1.2 j

• Now we have vector A and Vector B, we will now do the vector addition. We will add ith component of the A with ith component of B and Jth component of A with J^th component of B.

A+B = (1.23 +(-2.078) )i + (3.38 + (-1.2) )j

So the resultant vector  R= -0.848 i +2.18 j

• Here the  i^th component of the resultant vector R is negative and the J^thcomponent of the resultant vector R is positive. So this R vector lies in the 2nd quadrant.
• Now tanθ=Ry/Rx=2.18/(-0.848)= -2.5708

θ= tan^(-1) (-2.5708)= -68.744 degree .

Therefore it makes 68.744 degrees clockwise  with the negative X -axis.

Dot product :

The dot product is one way of multiplying two or more vectors. The resultant of the dot product of the vector is a scalar quantity, That’s why the dot product is also called a scalar product.

Suppose we have two vectors  a and b with components (a1, a2, a3 ) and (b1, b2, b3)

Then the dot product of vectors a and b will be given as

a∙b= a1*b1+ a2*b2+a3*b3

When the magnitudes of vectors and the angle between them is given then the dot product a∙b= |a||b| cosθ

A Properties of dot product:

• The dot product of two perpendicular vectors is zero. Two vectors are orthogonal only when a∙b=0

a∙b=|a||b|  cos⁡90= 0

• Dot product is commutative.   a∙b= b∙a
• Dot product is distributive.     a∙(b+c) = a∙b+a∙c
• Scalar multiple property :  (xa)∙(yb)= xy (a∙b)
• Since i, j and k are the unit vectors along the x, y and z-axis. These unit vectors are perpendicular to each other.

So , i∙j=j∙i=i∙k=k∙i= j∙k=k∙j=0

Also , i∙i=j∙j=k∙k=1

Let me show you this with an example

Cross product

The Vector product of the two vectors refers to a vector that is perpendicular to both of them. In other words, we can say that the cross-product of two vectors is a vector that is orthogonal to both.

If we have two vectors  a and b and the angle between them is θ

Then  c= a×b=|a||b| sinθ   n ̂   

When the two vectors a and b have components (ax, ay, az) and

 (bx, by, bz) respectively then a×b  will be given as

Let me show a solved example

Suppose we have two vectors x1 and x2 with components (2,-3,1) and (-2,1,1) respectively.

Properties of the cross product

• Cross product is anticommutative : a×b= -b×a.
• Cross product is distributive : a×(b+c)= a×b+ a×c.
• Cross product of two parallel vectors is zero.

a×b= |a||b| sinθ= |a||b|sin 0=0

• Cross product of two vectors is equal to the area of the parallelogram formed by two vectors.

• The direction of the cross product is given by the right-hand rule.

• Cross product of the unit vectors i, j and k will be given as.

• Also i×i=j×j=k×k=0
• Two vectors are parallel if their cross product is zero and vice versa.