Vectors

Two lines $\displaystyle L_1$ and $\displaystyle L_2$ are given by the equations:

$\displaystyle L_1: x = 4 + 3\lambda, y = 2 + 4\lambda, z = -7\lambda$
$\displaystyle L_2: \frac{x - 3}{-2} = \frac{y - 8}{1} = \frac{z + 1}{3}$

(a)Show that the lines $\displaystyle L_1$ and $\displaystyle L_2$ intersect and find the point of intersection.

(b)Calculate the obtuse angle between the lines $\displaystyle L_1$ and $\displaystyle L_2.$

Planes $\displaystyle \pi_1$, $\displaystyle \pi_2$ and $\displaystyle \pi_3$ have equations:

$\displaystyle \pi_1$: $\displaystyle x - 2y + z = -4$
$\displaystyle \pi_2$: $\displaystyle 3x - 5y - 2z = 1$
$\displaystyle \pi_3$: $\displaystyle -7x + 11y + az = -11$

where $\displaystyle a \in \mathbb{R}.$

(a)Use Gaussian elimination to find the value of $\displaystyle a$ such that the intersection of the planes $\displaystyle \pi_1$, $\displaystyle \pi_2$ and $\displaystyle \pi_3$ is a line.

(b)Find the equation of the line of intersection of the planes when $\displaystyle a$ takes this value.

The plane $\displaystyle \pi_4$ has equation $\displaystyle -9x + 15y + 6z = 20.$

(c)Find the acute angle between $\displaystyle \pi_1$ and $\displaystyle \pi_4.$

(d)Describe the geometrical relationship between $\displaystyle \pi_2$ and $\displaystyle \pi_4.$ Justify your answer.

The equations of two planes are given below.
$\displaystyle \pi_1: 2x - 3y - z = 9$
$\displaystyle \pi_2: x + y - 3z = 2$

(a)Verify that the line of intersection, $\displaystyle L_1$, of these two planes has parametric equations
$\displaystyle x = 2\lambda + 3$
$\displaystyle y = \lambda - 1$
$\displaystyle z = \lambda$

(b)Let $\displaystyle \pi_3$ be the plane with equation $\displaystyle -2x + 4y + 3z = 4.$
Calculate the acute angle between the line $\displaystyle L_1$ and the plane $\displaystyle \pi_3.$

(c)$\displaystyle L_2$ is the line perpendicular to $\displaystyle \pi_3$ passing through $\displaystyle P(1, 3, -2).$
Determine whether or not $\displaystyle L_1$ and $\displaystyle L_2$ intersect.