Assignment on Lines

1.

Find the vector and parametric equations for the line through the point $P=(2, 5, -1)$ and parallel to the vector $\v = \langle -3, 1, 2\rangle$.

2.

Find the vector and parametric equations for the line through the point $P=(5, 8, -6)$ and parallel to the vector $\v = 2\i -3\j + 4\k$.

3.

Find the vector and parametric equations for the line through the points $P=(4, 1, -8)$ and $Q = (2, 3, 5)$.

4.

Find the angle between the lines $l_1$ and $l_2$ given by:
$l_1: \r = \langle 1-2t, 3+t, 4 - 5t\rangle$ and $l_2: \r = \langle 2-s, 1-2s, 3+2s\rangle$.

5.

Find the parametric equations of the line through $(3, -1, 2)$ and parallel to the line $\r = \langle 2-3t, 7+t,8+5t\rangle$.

6.

Find the vector form of the line through the point $(5, 2, -3)$ and orthogonal to the lines $\r = \langle 2+t, 3-2t, 4 - 5t\rangle$ and $\r = \langle 1-t, 2t, 3+4t \rangle$.

7.

Determine whether the lines $l_1$ and $l_2$ are parallel, skew, or intersecting. If they intersect, find their point of intersection.

(a)

$l_1$: $x = 4 - t$, $y = 2t$, $z = 3 + 4t$, and $l_2$: $x = 2 + 3s$, $y = 1 - s$, $z = 4 + s$.

(b)

$l_2$: $\r = \langle 3 - 4t, 2 + t, 2t \rangle$, and $\r = \langle 3 + 2s, 1-s, 8 + 3s \rangle$