er=radial unit vector&et=tangential unit vector,now my question is how er=icostheta+jsintheta and et= -isintheta+jcostheta are derived? Please explain with diagram

To understand how the radial unit vector ${\mathbf{e}}_r$ and the tangential unit vector ${\mathbf{e}}_{\theta }$ are derived in polar coordinates, we can break down the concepts step by step. These vectors are essential in fields like physics and engineering, particularly when dealing with circular motion or any system that can be described in polar coordinates.

Understanding Polar Coordinates

In polar coordinates, a point in the plane is represented by its distance $r$ from the origin and an angle $\theta$ from the positive x-axis. The relationship between polar coordinates and Cartesian coordinates is given by:

• x = r * cos(θ)
• y = r * sin(θ)

Defining the Radial Unit Vector

The radial unit vector ${\mathbf{e}}_r$ points directly away from the origin towards the point in question. It is essentially the direction of increasing $r$. To express this vector in terms of its Cartesian components, we can use the relationships we mentioned earlier:

• In the x-direction: $\cos (\theta )$
• In the y-direction: $\sin (\theta )$

Thus, the radial unit vector can be expressed as:

${\mathbf{e}}_r=\mathbf{i}\cos (\theta )+\mathbf{j}\sin (\theta )$

Visualizing the Radial Unit Vector

Imagine a circle centered at the origin. As you move around the circle, the radial vector points outward from the center to the edge of the circle, making an angle $\theta$ with the positive x-axis. This vector's components are derived from the cosine and sine of that angle, which gives us the x and y components, respectively.

Exploring the Tangential Unit Vector

The tangential unit vector ${\mathbf{e}}_{\theta }$ is perpendicular to the radial vector and points in the direction of increasing $\theta$. To derive this vector, we can rotate the radial vector by 90 degrees counterclockwise. The new components can be found as follows:

• The x-component becomes $-\sin (\theta )$
• The y-component becomes $\cos (\theta )$

Thus, the tangential unit vector is given by:

${\mathbf{e}}_{\theta }=-\mathbf{i}\sin (\theta )+\mathbf{j}\cos (\theta )$

Visualizing the Tangential Unit Vector

Continuing with our circle analogy, the tangential vector at any point on the circle is directed along the edge of the circle, tangent to the path. It represents the direction in which the angle $\theta$ increases. This vector is crucial for understanding motion along a circular path, as it indicates the direction of velocity when an object moves along the circumference.

Summary of the Vectors

In summary, the radial and tangential unit vectors in polar coordinates are:

• ${\mathbf{e}}_r=\mathbf{i}\cos (\theta )+\mathbf{j}\sin (\theta )$ - points outward from the origin.
• ${\mathbf{e}}_{\theta }=-\mathbf{i}\sin (\theta )+\mathbf{j}\cos (\theta )$ - points in the direction of increasing angle.

These vectors are fundamental in analyzing motion in polar coordinates, allowing us to express forces, velocities, and other vector quantities in a way that aligns with the geometry of circular motion. Understanding their derivation helps in grasping more complex concepts in physics and engineering.