Lines in Space

Let $P_1 = (1,2,-1)$ and $P_2 = (-2,1,-2)$ and let $\mathcal{L}$ be the line in $\R^3$ through $P_1$ and $P_2$. Note that Figure shows a similar example of a line in 3D defined by two points.

1. Find a direction vector for the line $\mathcal{L}$.

2. Find a vector equation of $\mathcal{L}$ in the form $\vr(t) = \vr_0 + t\vv$.

3. Consider the vector equation $\vs(t) = \langle -5, 0, -3 \rangle + t \langle 6, 2, 2 \rangle.$ What is the direction of the line given by $\vs(t)$? Is this new line parallel to line $\mathcal{L}$?

4. Do $\vr(t)$ and $\vs(t)$ represent the same line, $\mathcal{L}$? Write a couple of sentences to j…

Let $P_1 = (1,2,-1)$ and $P_2 = (-2,1,-2)$, and let $\mathcal{L}$ be the line in $\R^3$ through $P_1$ and $P_2$. (This is the same line as in Activity.)

1. Find parametric equations of the line $\mathcal{L}$.

2. Does the point $(1, 2, 1)$ lie on $\mathcal{L}$? If so, what value of $t$ results in this point?

3. Consider another line, $\mathcal{K}$, whose parametric equations are What is the direction of the line $\mathcal{K}$?

4. Do the lines $\mathcal{L}$ and $\mathcal{K}$ intersect? If so, provide the point of intersection and the $t$ and $s$ values, r…

Let $P_1 = (1,2,-1)$ and $P_2 = (-2,1,-2)$, and let $\mathcal{L}$ be the line in $\R^3$ through $P_1$ and $P_2$ as in Activity.

1. What value of the parameter $t$ makes $(x(t), y(t), z(t)) = P_1$? What value of $t$ makes $(x(t), y(t), z(t)) = P_2$?

2. What $t$ values describe the line segment between the points $P_1$ and $P_2$?

3. What about the line segment (along the same line) from $(7,4,1)$ to $(-8,-1,-4)$?

4. Now, consider a segment that lies on a diffe…