Why is the direction of angle theta in circular motion towards and inwards?

Question

Why is the direction of angle $\theta$ in circular motion towards and inwards of plane x-y axis?

I am not getting this concept at all.As the angel theta is changing ; the arc length (s in diagram ) is also changing;

We know about vector cross product of two vectors a and b whose direction is perpendicular to the plane of paper( or perpendicular xy axis on my paper i.e xz axis).So my sir tells me that to find direction of angle θ (it is on a 2D plane, X-Y axis), then its direction is on X-Z plane. (You can interpret the direction of vector cross product by right hand thumb rule ).

Also, that z axis is drawn from the centre of circle. Only from there only.

Answer 1

This answer is based on the comment that the statements in the question occurred during instruction in physics.

There is a convention in physics of using a vector to represent rotation around a fixed line in space (the "axis of rotation", which may or may not also be a coordinate axis). The length of the vector is proportional to the amount of rotation. This convention has some advantages when representing rotational speed, but in introductory physics it might also be used to represent the total angle through which the rotation was performed. If you rotate twice as much, you make the vector twice as long.

If you have an object traveling in a circle around the origin in the $x,y$ plane, this is a rotation around the $z$ axis. To represent a rotation through a certain angle, which moves the object a certain distance around the circle, you make a vector in the same direction as the $z$ axis and make its length equal to the rotation measured in radians (equal to the distance traveled around the circle if the circle has radius $1.$)

To decide whether the vector should point in the positive $z$ direction or the negative $z$ direction, you use another convention, usually a "right hand" rule. The idea that the vector has to "start" in the center of the circle is your instructor's convention, perhaps to help you remember graphically where the axis of rotation is. For a mathematician, a three-dimensional vector in three-dimensional space does not start anywhere: it merely has a direction and a length.

This particular use of a rotation vector has limited usefulness. In special cases you can represent the result of doing one rotation and then another rotation by adding the rotation vectors, but only if the rotations are around the exact same axis (or at least around parallel axes). The triangle rule for adding vectors gives wrong answers to this kind of problem.

Answer 2

The direction of $\theta$ being positive moving counterclockwise in the $xy$-plane is typically an assumption that we use along with the right-hand rule to determine the direction of the normal vector for the surface enclosed by a closed path, a circle specifically in this question.

Given a circle in the $xy$-plane, with the positive direction of $\theta$ being counterclockwise along that circle, we could theoretically define the direction of the normal vector either in the positive $z$ direction or the negative $z$ direction. Either way, the vector would be normal since both directions are orthogonal to any vector in the $xy$-plane.

By convention, we define the positive $z$-direction using the right-hand rule, and we use the same convention to define the positive direction of the normal vector for our surface. This is also the same convention that gives us $\mathbf{\hat{i}} \times \mathbf{\hat{j}} = \mathbf{\hat{k}}$, where $\mathbf{\hat{i}}$ is the unit vector in the $x$-direction, $\mathbf{\hat{j}}$ is the unit vector in the $y$-direction, and $\mathbf{\hat{k}}$ is the unit vector in the $z$-direction. This is all to ensure that "positive" means the same thing in slightly different contexts (though the underlying mathematics is the same).