How do I write the Symmetric Equation of a line given the following parametric equations: (where t is a scalar)
x=0t
y=0t
z=1+1t
I know normally I would solve each equation for t and set the equations equal to one another. However, I can not solve for t in the first two equations. What do I do? Thanks for your help.
To write the symmetric equation of a line given the parametric equations, you can eliminate the parameter by expressing one variable in terms of the other variables.
In this case, we can start by getting rid of the parameter 't'.
From the first equation, x = 0t, we can see that x is always zero for any value of t.
From the second equation, y = 0t, we can see that y is also always zero for any value of t.
Now, we can substitute these values of x and y into the third equation:
z = 1 + 1t
Since x and y are both zero, we have:
z = 1 + 1t
So, the symmetric equation of the line is:
x = 0
y = 0
z = 1 + t
The line is parallel to the z-axis and passes through the point (0, 0, 1).
To write the symmetric equation of a line, you usually need the direction ratios of the line. However, in this case where the parametric equations for the line are x = 0t, y = 0t, and z = 1 + 1t, these equations already provide you with the direction ratios.
The direction ratios for x, y, and z are 0, 0, and 1, respectively. Using these direction ratios, you can form the symmetric equation of the line.
The symmetric equation of a line in three-dimensional space has the form:
(x - x0) / a = (y - y0) / b = (z - z0) / c
where (x0, y0, z0) is a point on the line and a, b, and c are the direction ratios of the line.
Since the direction ratios are 0, 0, and 1 in this case, we can choose any point on the line to substitute for (x0, y0, z0). For simplicity, we can use the point (0, 0, 1) since it satisfies the given parametric equations.
Substituting the values, we get:
(x - 0) / 0 = (y - 0) / 0 = (z - 1) / 1
Simplifying, we have:
x / 0 = y / 0 = z - 1
Since dividing by zero is undefined, we can rewrite the equation as follows:
x = 0
y = 0
z - 1 = 0
Therefore, the symmetric equation of the line is:
x = 0
y = 0
z = 1
This represents a line parallel to the z-axis, passing through the point (0, 0, 1).