Write parametric and symmetric equation for the z-axis

[NATURE.M]

Homework Statement

Write parametric and symmetric equations for the z-axis.

Homework Equations

vector, parametric and symmetric equations, in general form.

The Attempt at a Solution

I believe I have obtained the correct answer, would just like confirmation.
Let our direction vector be b=[0,0,1], and a point on z axis be A(0,0,0).

Vector equation:
[x,y,z]=[0,0,0]+t[0,0,1], tεR

Parametric equations:
x=0 y=0 z=t

Symmetric equations:
t=z ; x=0 ; y=0

 

[Mark44]

Homework Statement

Write parametric and symmetric equations for the z-axis.

Homework Equations

vector, parametric and symmetric equations, in general form.

The Attempt at a Solution

I believe I have obtained the correct answer, would just like confirmation.
Let our direction vector be b=[0,0,1], and a point on z axis be A(0,0,0).

Vector equation:
[x,y,z]=[0,0,0]+t[0,0,1], tεR

Parametric equations:
x=0 y=0 z=t

Symmetric equations:
t=z ; x=0 ; y=0

The parametric equations look fine, although you might add that t
$\in$ R.

Your symmetric equations aren't correct. As I recall, they should look like this:

$$ \frac{x - a}{A} = \frac{y - b}{B} = \frac{z - c}{C} $$
Here (a, b, c) is a point on the line and <A, B, C> is a vector that is parallel to the line. If any component of the direction vector happens to be zero, it's possible to have a corresponding 0 in the denominator in the form above.

 

[NATURE.M]

When the denominator is 0, I was told the symmetric equations are not possible. But can be expressed in the unusual form that I provided. I've looked in one calculus textbook and a separate lesson sheet that indicate the method that I used. The separate lesson sheet can be found at, http://www.jackmathsolutions.com/images/Ch_5_Lines.pdf, on page 121.
So is this source wrong then?

 

[Dick]

I would be tempted to write the symmetric form as x=y=0*z.

 

[Mark44]

It appears that authors are not in agreement on this. One of the calculus books I have (Calculus and Analytic Geometry, 2nd Ed., Abraham Schwartz, p.590) has this to say about the symmetric form of a line (emphasis added):

Remark 2
If one of the components of v is 0, then one of the members of the symmetric-form equations for a line L in v's direction will have 0 as its denominator. In such a case, the symmetric-form description of L is to be interpreted as a statement about proportional trios of numbers.

Obviously, we never allow division by zero in mathematics, so having a zero in the denominator does not in any way imply a division operation. In such cases, the symmetric equations are more along the lines of notation.

So as not to have to educate your instructor (who wrote the notes you posted?) it's probably a good idea to follow the form in those notes. This would make your symmetric equations look like this:

x = 0, y = 0, (z - 1)/1 = t

 

[NATURE.M]

okay, and I assume the calculus text your referencing is of a higher level in contrast to the one I'm using.
So then in that context, x=0; y=0; (z)/1=t would be the proper symmetric equations for the question I provided.

 

[Mark44]

okay, and I assume the calculus text your referencing is of a higher level in contrast to the one I'm using.

No, that was a standard calculus text that was used in the '60s.

So then in that context, x=0; y=0; (z)/1=t would be the proper symmetric equations for the question I provided.

 

[NATURE.M]

Mark44 do you think it would be better if I just simply stated its not possible for the symmetric equations. After looking at them,x=0; y=0; (z)/1=t , there exactly identical to the parametric equations, so it seems silly to write them.

 

[Mark44]

According to the book you cited, symmetric equations aren't possible, but I cited a well-known text that shows that they are. Also, Dick gave another way that they could be written.

 

[NATURE.M]

According to the book you cited, symmetric equations aren't possible, but I cited a well-known text that shows that they are. Also, Dick gave another way that they could be written.

Okay, I'll probably stick to the 'not possible' answer then, since its probably the simplest answer.

 

[HallsofIvy]

I would say that the symmetric equations are x= y= 0. The fact that there is no "z" in that is what says that z can be any number.