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 x = 0t, y = 0t, and z = 1 + 1t, you can follow these steps:
1. Start by writing the parametric equations in vector form:
r = <0t, 0t, 1 + 1t>
2. Notice that x = 0t and y = 0t can be simplified to x = 0 and y = 0, respectively.
3. Now, the vector form of the line becomes:
r = <0, 0, 1 + 1t>
4. To write the symmetric equation, we express each component of the vector separately. Therefore, we have:
x = 0
y = 0
z = 1 + 1t
5. Simplifying, we can rewrite the symmetric equation as:
x = 0
y = 0
z - 1 = t
Therefore, the symmetric equation for the given line is:
x = 0, y = 0, and z - 1 = t.
To write the symmetric equation of a line given its parametric equations, you would need to eliminate the parameter t. In this case, since you cannot solve for t in the first two equations, you can take a different approach.
Instead of trying to solve for t, you can use the fact that if two quantities are equal to a third quantity, then they are equal to each other. So, we can set x equal to y and solve for t.
Since x = 0t and y = 0t in this case, we have 0t = 0t. This equation is true for all values of t. Hence, we can say that x = y and eliminate the parameter t.
Now we have two equations: x = y and z = 1 + 1t.
To write the symmetric equation, we need to express x, y, and z in terms of a single variable, let's call it s.
Since x = y, we can replace y with x in the equation z = 1 + 1t:
z = 1 + 1t becomes z = 1 + 1x.
Now, we have x = y and z = 1 + x.
To write the symmetric equation, we can rearrange these equations to isolate x, y, and z:
x - y = 0 (subtracting y from both sides of x = y)
z - x = 1 (subtracting x from both sides of z = 1 + x)
Now we can express the symmetric equation as:
x - y = 0 and z - x = 1
This equation represents the line in symmetric form, where the coefficients of x, y, and z represent the direction ratios of the line.