Can someone help me with this please.

A vector field in cylindrical polar co-ordinates is given by
V = R Rˆ + R 3 s i n ( φ ) c o s ( φ ) φˆ + 3 z kˆ

where Rˆ, φˆ, kˆ are the appropriate unit vectors. Translate this vector field into the
Cartesian x, y, z co-ordinate system.

What is the first step??

Could you please guide me through the steps.
Thanks.

I get that
x = r cosφ
y = r sinφ
z = z

And i know i need to sub 'unit R^' with cosφ i + sinφ j and z with 'unit K^' with z k.

How about 'unit φ^' ?
What do i need to sub phi with?
I look else where it says sub it with -sinφ i + cosφ j.
But how? and why?
Can someone explain it to me?

Thanks alot!

In your previous post, you have been asked what R3 stands for, and you have not clarified. We do not know yet if 3 is a superscript, a subscript, or just a number written in the wrong place.

Yes, φ equals a vector
< -sinφ, cosφ, 0 >.
Also, are
R,φ and z constants?
How is k related to z, there must exist a multiplicative constant, unless the constant equals 1.

Take a look at your previous post. See whether my solution makes any sense, and whether I interpreted things correctly.

To translate the vector field from cylindrical polar coordinates to Cartesian coordinates, you are correct that you need to substitute the appropriate unit vectors. In this case, you already know how to substitute the unit vectors for R^ and K^, but you are unsure about how to substitute the unit vector for φ^.

The unit vector for φ^ can be determined by considering the direction in which the angle φ increases. In cylindrical polar coordinates, φ increases in the counterclockwise direction when viewed from above the z-axis. To determine the coordinates of the unit vector for φ^, imagine a small displacement in the positive direction of φ. We can observe that this displacement will be along the tangent to a circle of radius R at a given point. The tangent to the circle will be perpendicular to the radius vector pointing towards the center of the circle. Therefore, the unit vector for φ^ should be perpendicular to R^.

Since R^ is given by cos(φ)i + sin(φ)j (as you correctly mentioned), a vector that is perpendicular to it can be obtained by rotating R^ by 90 degrees counterclockwise. This rotation can be achieved by multiplying R^ by the rotation matrix:

┌ ┐
│ 0 -1 │
│ │
│ 1 0 │
└ ┘

So, multiplying the rotation matrix with R^ gives:

-sin(φ)i + cos(φ)j

This is why you need to substitute the unit vector for φ^ with -sin(φ)i + cos(φ)j. It represents the direction perpendicular to R^.

I hope this clarifies the concept of substituting the unit vector for φ^. If you have any further questions, feel free to ask!