Find a line that lies entirely in the set defined by the equation x2 + y2 − z2 = 1.

We want to find a line that lies in the set defined by x^2 + y^2 - z^2 = 1

That line can be <1, t, t>.

So we want to have a line that is defined by the given restriction.

We can rewrite the restriction as:

x^2 + (y^2 - z^2) = 1

Now we can add another restriction (just to find the line) that is defining:

y = z (notice that if we fix the value of x, this will define a line).

Then we will have:

x^2 + (y^2 - y^2) = 1

x^2 = 1

Now we can fix a value for x, for example x = 1

Then our line is defined by:

y = z and x = 1.

Writing it in vector form we would have something like:

<1, t, t>

Replacing that in the restriction, we get

1^2 + t^2 - t^2 = 1 + 0 = 1

As expected.

If you want to learn more, you can read:

link

Step-by-step explanation:

A line is a one-dimensional figure that has no thickness and extends infinitely in both directions.

A line segment is a line that has two end-points and a ray is a line that has one end-point.

Given equation is

To find : a line that lies entirely in the set defined by the given equation.

Solution:

Take

Check:

Therefore, satisfy the given equation.

"image"

step-by-step explanation:

in most of these problems, the notation is that the point with a "prime" is the image of the point without. that is, b' is the image of b.

here, there is no dilation or reflection. there is simply translation (with no direction shown). which figure you call the "preimage" and which one you call the "image" is up to you. as a matter of convention, the naming is as described above.