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| This has the side effect of associating a direction with movement along the line. At ''t=0'', the solution is (0,0). At ''t=1'', the solution is (-1,1). Applications of parametric equations then have three parts: * ''does'' a line pass through some point, line, or plane? * ''when'' does it pass through? * ''where'' does it pass through? |
This has the side effect of associating a direction with movement along the line. At ''t=0'', the solution is (0,0). At ''t=1'', the solution is (-1,1). |
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| Given a system described by ''x=2t-1'', ''y=t+2'', and ''z=-3t+2'' and a plane ''x+2y+4z=7'', the first and second questions are answered by substitution. | Especially with multiple variables, parametric equations are sometimes re-expressed in several ways: * This triplet... * x = 1+2t * y = 3-t * z = 5t * ...is equivalent to this symmetric equation... * (x-1)/2 = (y-3)/-1 = z/5 * ...and is also equivalent to this vector form... * ''[1, 3, 0] + t[2, -1, 5] = [1, 3, 0] + [2t, -t, 5t] = [1+2t, 3-t, 5t]'' * ...and can therefore be expressed as a '''vector-valued function'''... * ''r(t) = f(t)i + g(t)j + h(t)k = (1+2t)i + (3-t)j + (5t)k'' |
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| ''(2t-1)+2(t+2)+4(-3t+2)≟7'' | ---- |
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| ''2t-1+2t+4-12t+8≟7'' | |
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| ''-8t+11≟7'' | |
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| ''-8t≟-4'' | == Usage == |
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| ''t=.5'' | A parametric equation is useful for characterizing intersections, e.g. between the parameterized line and a given point or plane. |
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| If there was no solution to satisfy equality, then the line must not pass through. (The third question is then answered by substituting this solution into the original system.) | There is also a straightforward method for calculating arc length of a parametric equation. Given a vector-valued function ''r(t) = f(t)i + g(t)j + h(t)k'': |
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| Given a vector a⃗ and a starting point A,,0,, (e.g., the origin), a parametric equation can be expressed as ''A,,t,, = A,,0,, + a⃗t''. The same can be said when given two points in ''R^3^'' as ''A,,0,,'' and ''A,,1,,''. These trivially yield a vector a⃗ as ''[A,,1x,,-A,,0x,, A,,1y,,-A,,0y,, A,,1z,,-A,,0z,,]''. | {{attachment:arc.svg}} |
Parametric Equation
A parametric equation is a reformulation of an equations in terms of time, such that there is now a direction associated with movement along the equation.
Contents
Description
A line can often be described with a single equation. This below line, however, requires two equation to be expressed: y=√(x+1)+1 and y=-√(x+1)+1.
This can be addressed by reformulating the equation in terms of time t: y=f(t) and x=g(t). In this specific example, the parametric equations are y=t and x=(t-1)2-1.
This has the side effect of associating a direction with movement along the line. At t=0, the solution is (0,0). At t=1, the solution is (-1,1).
Especially with multiple variables, parametric equations are sometimes re-expressed in several ways:
- This triplet...
- x = 1+2t
- y = 3-t
- z = 5t
- ...is equivalent to this symmetric equation...
- (x-1)/2 = (y-3)/-1 = z/5
- ...and is also equivalent to this vector form...
[1, 3, 0] + t[2, -1, 5] = [1, 3, 0] + [2t, -t, 5t] = [1+2t, 3-t, 5t]
...and can therefore be expressed as a vector-valued function...
r(t) = f(t)i + g(t)j + h(t)k = (1+2t)i + (3-t)j + (5t)k
Usage
A parametric equation is useful for characterizing intersections, e.g. between the parameterized line and a given point or plane.
There is also a straightforward method for calculating arc length of a parametric equation. Given a vector-valued function r(t) = f(t)i + g(t)j + h(t)k:
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