1. ## Vector Equations

2 questions

1. Determine the value of a for which the following planes intersect in a line:

x-2y-z=0
x+9y-5z=0
ax-y+z=0

2. Given the Scaler Equation of a plane, determine a corresponding vectot equation. 2x+5y-z+10=0

Lastly any good school books on vectors?

2. Originally Posted by someone21
2 questions

1. Determine the value of a for which the following planes intersect in a line:

x-2y-z=0
x+9y-5z=0
ax-y+z=0

2. Given the Scaler Equation of a plane, determine a corresponding vectot equation. 2x+5y-z+10=0

Lastly any good school books on vectors?
to #1:

If
$p_1: x-2y-z=0$ then the normal vector of $p_1$ is $\overrightarrow{n_1} = (1, -2, -1)$

$p_2: x+9y-5z=0$ then the normal vector of $p_2$ is $\overrightarrow{n_2} = (1, 9, -5)$

$p_3: ax-y+z=0$ then the normal vector of $p_3$ is $\overrightarrow{n_3} = (a, -1, 1)$

The direction vector of the intersection line of the first two planes is:

$\overrightarrow{n_1} \times \overrightarrow{n_2}=(19, 4, 11)$

The normal vector $\overrightarrow{n_3}$ must be perpendicular to this direction vector:

$\overrightarrow{n_3} \cdot (19, 4, 11) = 0~\implies~ (a, -1, 1) \cdot (19, 4, 11) = 0$

which will yield $a = -\frac7{19}$

to #2:

Given
$2x+5y-z+10=0$
That means:
$z = 2x+5y+10$

Now substitute x = r and y = s

$\begin{array}{l}x = r \\ y = s \\ z = 10 +2r+5s\end{array}$ ...... $\implies$ ...... $(x, y, z) = (0, 0, 10) + r(1, 0 , 2) + s( 0, 1, 5)$