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Mth 277
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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I guess I could benefit from a little clarification on going from the from v = i + j + k to parametric form of a line in R^3
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I get that Parametric Form of line is...
x = x_0 + tA, y = y_0 + tB....
Where A, B, and C are coefficients of the i, j and k terms of the vector. But I still have trouble solving some of the questions about Parametric equations and such because of the conversion from one to thee other and back. Maybe if I had a better understanding of how one relates to the other then it would not be so confusing.
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@& A quick summary:
If you start out at point (x0, y0, z0) and move at velocity v = a `i + b `j + c `k for t seconds (where `i, `j and `k are the unit vectors in the x, y and z directions), then your displacement will be a t `i + b t `j + c t `k, and your position will be (x0 + t a, y0 + t b, z0 + t c).
Now you're moving along a straight line, and the line is in the direction of vector v, so all the points (x0 + t a, y0 + t b, z0 + t c) lie on a line through (x0, y0, z0) in the direction of v. And since t is a continuous variable that can take any positive or negative value, every point on the line can be expressed in this form.
So (x0 + t a, y0 + t b, z0 + t c) is the general representation of a point on the line through (x0, y0, z0) in direction v.
This is equivalent to saying that
x = x0 + a t
y = y0 + b t
z = z0 + c t.
These are parametric equations for the line, in terms of the parameter c. Again, as our parameter varies over all real values, our equations give us all points on the line, and only points on the line.
If we solve these three equations for t we get
t = (x - x0) / a
t = (y - y0) / b
t = (z - z0) / c
so that
(x - x0) / a = (y - y0) / b = (z - z0) / c.
These are the symmetric equations of the line.
Any point (x, y, z) whose coordinates satisfy these equations is on the line. And the coordinates of every point on the line satisfy these equations.
Another note: The Class Notes were incorrectly linked to the wrong folder. I've fixed the link.
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