Motion in Plane

Resolution of Vectors:

A given vector can be expressed as sum of two vectors. Let us assume there are two non-zero vectors a and b, and there is a third vector A. All of them are lying in a plane; as shown by given figure. Vector A is shown by OP. Through O, draw a straight line parallel to a. Similarly, through P draw a straight line parallel to b. These new lines intersect at Q.

A = OP = OQ + QP

Since OQ = λa and QP = μb

Where λ and μ are real numbers.

So, A = λa + μb

Thus, we can say that A has been resolved into two component vectors λa and μb along a set of two vectors: all the three lie in the same plane.

Unit Vector: A vector of unit magnitude is called unit vector. Unit vector has no dimension and unit, rather it is used to specify a direction only. Unit vectors along the x-, y- and z-axes of a rectangular coordinate system are denoted by î, ĵ and k̂ respectively.

|î|=|ĵ|=|k̂| = 1

We can resolve a vector A in terms of component vectors that lie along unit vectors. Let us take a vector A which lies in x-y plane. Draw lines from head of A perpendicular to the coordinate axes to get vectors A₁ and A₂ so that A₁ + A₂ = A.

Since A₁ is parallel to î and A₂ is parallel to ĵ hence we have

A₁ = A_x î

A₂ = A_y ĵ

Where A_x and A_y are real numbers.

Now, A_x = A cos θ

A_y = A sin θ

From above equations, it is clear that component of a vector can be positive, negative or zero, and it depends on magnitude of A and on angle θ it makes with x-axis.

If A and θ are given A_x and A_y can be obtained using above equations. If A_x and A_y are given then A and θ can be found as follows:

A_x² + A_y² = A² cos² θ + A² sin² θ = A²

Or, `A=sqrt(A_x^2+A_y^2)`

And, tan θ =`(A_y)/(a_x)`

Or, θ = tan^-1 `(A_y)/(A_x)`

Following equations show resolution of vectors in case of three axes and three unit vectors.

A_x = A cos α

A_y = A cos β

A_z = A cos γ

A = A_x î + A_y ĵ + A_z k̂

`A=sqrt(A_x^2+A_y^2+A_z^2)`

Position vector r can be expressed as

r= xî = yĵ + zk̂

Where x, y and z are the components of r along x-,y- and z-axes respectively.

Vector Addition:
Analytical Method

Let us take two vectors A and B in x-y plane with components A_x, A_y and B_x, B_y

A = A_xî + A_yĵ

B = B_xî + B_yĵ

If R is the sum of given vectors then:

R = A + B

= (A_xî + A_yĵ) + (B_xî + B_yĵ)

Since vectors obey the commutative and associative laws, above equation can be written as follows:

R =(A_x + B_x)î + (A_y + B_y)ĵ

Since R = R_xî + R_yĵ

So, R_x = A_x + B_x

And R_y = A_y + B_y

This shows that each component of the resultant vector R is the sum of the corresponding components of A and B. This method can be extended in three dimensions as well and also to addition and subtraction of any number of vectors.

Position Vector and Displacement

Let us assume that a particle P is located in a plane with reference to origin of an x-y reference frame. The position vector r of particle P is given by following equation:

r = xî + yĵ

Where x and y are components of r along x- and y-axes. We can also say that x and y are coordinates of the object.

Let us assume that a particle moves from P to P' between time t to t'. Then displacement can be given by following equation:

Δr = r' - r

This equation can be written in component form as follows:

Δ r = (x'i + y'ĵ) – (xî + xĵ)

= î(x' - x) + ĵ(y' - y)

= îΔx + ĵΔy

Motion in Plane

Resolution of Vectors:

Vector Addition: Analytical Method

Position Vector and Displacement

Vector Addition:
Analytical Method