question regarding section 10.5

Professor,

I’m wondering about an equation, equation 4, given on page 569 of our textbook. Do these statements show the right relationship?

    

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The first equation is fine, it describes the line segment starting at the vector (point) r_0  and passing through to the endpoint vector (point) r_1, because r(0)=r_0 and r(1)=r_1.  Note that this equation can be rewritten as 

r(t) = (1-t)r_0 + t r_1 

      = r_0 - t r_0 + t r_1 

      = r_0 + t (r_1 - r_0)   

      = r_0 + t v   

where v= (r_1 - r_0), which is just the form of a line as we described today in lecture, and allows all real values of t, i.e. does not require 0≤t≤1.

I'm not sure what the second equation means, since I'm not sure what you mean by r_a and r_b, but it would give some line segment not necessarily beginning or ending at  r_a and r_b/

(Edit the morning after: after thinking a bit more about your question I think I understand where you were coming from: you wanted to know if 0 and 1 are special or if any numbers would do.  The answer is no, any numbers will not do and 0 and 1 are special.  The whole point is that

(1-t)u+t v 

interpolates between u and v as t goes from 0 to 1, no matter what vectors u and v are.)