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Sketch the line segment represented by each vector equation:

$$\begin{align} r &= (1-t)(i+j) + tk \;& 0 \le t \le 1 \\ \\ r &= (1-t)(i+j+k) + t(i+j) \;& 0 \le t \le 1 \end{align}$$

I'm having a really hard time trying to figure this out. I'm in my textbook and the only thing it has on this topic is

$r = r_{0} + t(r_{1} - r_{0})$ or $r = (1-t)r_{0} + tr_{1}$

It doesn't give any examples on how to graph the line segment. (I've even looked in the back of the book on previous similar problems not assigned to me, and they don't give any clues on why they are graphed the way they are...)

Could someone help explain this to me?

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Since it is difficult to "sketch" in three dimensions, for (b) you will have to make it clear that the plane of the paper you are using for your sketch is the vertical plane that contains the point $(1,1,0)$, or equivalently the vector $i+j$. Then you will label one point $(1,1,0)$ (or $i+j$), another point $(1,1,1)$ (or $i+j+k$), and draw the line segment that joins these two points. –  André Nicolas Feb 23 '12 at 3:16

1 Answer 1

up vote 1 down vote accepted

Plug in $t=0$ and $t=1$ to get the two endpoints of the line segment. This works because

$$\mathbf{x}(t)=(1-t)\mathbf{a}+t\mathbf{b}=\mathbf{a}+(\mathbf{b}-\mathbf{a})t$$

starts at $\mathbf{a}$ when $t=0$ and will go in a straight $\mathbf{b}-\mathbf{a}$ direction towards $\mathbf{b}$ (when $t=1$).

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