By design, LineaAlgebra works over the complex number field. Hence, variables are assumed to be complex, and hence the conjugation mentioned by Carl Love. Within this package, some commands allow for an option to treat the field as real.

The Student LinearAlgebra package works over the real number field and the need for constantly dealing with complex conjugates disappears. Make the choice that's best for your needs.

If you are working in the Student package and find you need something that's only in the LinearAlgebra package itself, use the long name for the command, as in LinearAlgebra:-DotProduct.

@ecterrab 

I often use commands from the Vector Calculus and Multivariate Calculus packages where there are name clashes. I've gotten around this problem by aliasing a short name to the long name of one of the packages and not loading that package. For example, 

alias(VC = Student:-VectorCalculus):

Then (for example) VC:-PlotVector(...)

Equivalent constructs appear in a number of my Maplesoft webinars. However, I've developed no worksheets in which both VectorCalculus and Physics:-Vectors are used, so I have no actual experience in using both these two packages simultaneously.

@The function 

The help systems in either Maple 2021 or 2022 contain the Calculus Study Guide and the Multivariate Calculus Study Guide. These are ebooks that demonstrate how to solve more than 1200 exercises in the calculus using the tools of Maple. Are you aware of these resouces?

@tomleslie 

Thanks for the extended exposition. My aim for my earlier response was simply to call attention to the paradigm: Use the most direct set of tools available in Maple. Apparently, the OP might not have been familiar with the tools in the VectorCalculus packages. To my mind, knowledge of such tools allows for a more conceptual thinking about mathematics.

@JAMET 

The ConicsTutor in the Student Precalculus package will graph and interpret a conic section not given in standard form. The results are reported in floating-point form, and no computational details are provided.

This tutor is available from the Tools/Precalculus menu as "Conic Sections" or by executing the command Student:-Precalculus:-ConicsTutor().

@ianmccr 

Yes, the Atomic Variables will look like ordinary Maple notation (i.e., not magenta) by simply unchecking "Atomic Variables" in the View menu.

@vv 

A simple extension, nicely implemented. Thanks.

@vv 

The "vertical" line is not an asymptote. It is an artifact of Maple's connect-the-dots plotting algorithm. It looks like an asymptote, but it really isn't. There are ways to suppress it, and there are Maple commands and tools to draw rational functions and their asymptotes, both vertical and horizontal. I suspect you probably know all this, but for the sake of anyone else reading your comment, I thought a clarification might be in order.

@Carl Love 

I initially passed on this question but Carl's post provokes me to respond.

If w=f(x,y,z), then w=0 and w=1 are two level surfaces defined, as in the example function provided, implicitly. An earlier response says to use implicitplot3d. Yes.

If in the same breath we are asked about level curves on a function of two variables, then that breath should not speak of f(x,y), but rather, should use a different function name. There can be no logical connection between the level surfaces of f(x,y,z) and the level curves of f(x,y). Doesn't make much sense to me, and I would give this same lecture to any student who posed the question asked by the OP using the same confusion between the name "f" for a function of three variables and for a function of two variables.

The level curves of g(x,y) are obtained in Maple with the implicitplot command (or even the contourplot command), as pointed out by tomleslie.

@VK057 

Sorry about that. It's not the letter V, but the keystrokes for a shilled fraction. Maple typesets a fraction as a-over-b, but to get fractions to take just 1 horizontal line, as in a/b, you need to escape the forward (/) slash. The escape character is the backslash (\). Hence, when put together, it looks like the letter V, but that appeared only because I copied/pasted some typeset 2D math, and it pasted in the two characters instead of just the one (/).

(I learned "shilled fraction" from the editor of my Advanced Engineering Math book more than 2 decades ago. I pretty much write all my fractions in Maple that way.)

I should have looked at what I pasted into my response before I sent it off.

@bstuan 

The usage is d1:=Distance(..., not d1:=distance(...

Maple is a symbolic language. The line L0 is held in Maple's memory with the unknown parameters a,b,c, and the distance between the lines is an expression containing those unknowns.

Thus, d1 and d2 are two expressions both containing a,b,c as unknowns. Maple's solve command does not insist on explicitly setting an expression to zero. It assumes the "right-hand-side" is to be take as zero. Hence, when solve is applied to d1 and d2, the values for a, b, c that make the two distances zero are computed.

Does that help?

@mmcdara 

There's a third way: the PlotVector command in the VectorCalculus packages.

The advantage of this command: Simply define the vector as a vector or as a RootedVector, and hit it with the PlotVector command. The two arrow commands in Maple require a lot of re-writing of the vector to conform to the differing syntax of these two commands.

@acer 

The TangentLine command in the VectorCalculus packages accepts the equation of the curve in implicit form, obviating the need to solve for y=f(x).

@janhardo 

(If indeed it is an objection.) It takes 4 keys to make one of these conversions: ALT, R, V, and one of 2, M, or i.

Math can be entered in one of 3 ways, and for each method of entry, there are two other forms. Hence, there are 6 transformations possible. How can this be done with one key?

@Carl Love 

The Context Panel's "simplify" includes the option "With Side Relations." A pop-up dialog then has a place for the side relation.

Hence, It takes just two visits (simplify and solve) to the Context Panel to implement the equivalent of the coded solution.