Symmetric to vector equation.How do you change a symmetric equation to a vector equation? (e.g`(2-x)/2=y=1+z` into `v=(2,3,1)+t(1,2,3)` ?

Expert Answers
txmedteach eNotes educator| Certified Educator

In any case where we have a situation with three variables and we want to parameterize to make a vector equation, it is almost always easiest to just set one of the variables to the main parameter.

Let's take the case you used for your example:

`(2-x)/2 = y = 1+z`

In this case, y is already isolated in terms of z and x, so let's just let t = y. Now, let's figure out what x and z are in terms of t:

`(2-x)/2 = t`

`1+z = t`

Solving both equations yields:

`x = 2-2t`

`z = t-1`

Now, we can get a full vector equation recognizing the following relations with the parameter t:

`x = 2-2t`

`y = t`

`z = t-1`

Our vector equation will be the same thing, except that it will separate the constant components and the coefficients of t:

`vecF(t) = (2,0,-1) + t(-2,1,1)`

I hope that helped explain the situation. You could have set t equal to x or z, too. The result would be the same line.