Vtyjkyuk

Find the value of $a$ for which the three planes
beginalign*
Pi_1 colon phantom2 x - 2y + z &= 7 \
Pi_2 colon 2x + y - 3z &= 9 \
Pi_3 colon phantom2 x + y - az &= 3
endalign*
do not intersect.

(Original image here.)

I'm attempting to solve this problem. I tried solving these equations simultaneously and arrived at $a= 1/z + 2$. I don't know how to continue or if what I've been doing is even correct.

Some help would be greatly appreciated.

Working:

$x-2y+z=7$

$2x+y-3z=9$

$x+y-az=3$

$4x+2y-6z=18$

$5x-5z=25$

$2x+2y-2az=6$

$3x-2az+z=13$

$15x-15z=75$

$15x-10az+5z=65$

$-15z+10az-5z=10$

$-20z+10az=10$

$2-a=-1/z$

$a=1/z+2$

Guide:

• We do not want the columns to span $mathbbR^3$ or it will be consistent.




• Let $C_i$ be the $i$-th column.




• If the third column is a linear combination of the first two, determine $a_1$ and $a_2$ where $C_3=a_1C_1+a_2C_2$ using the first two rows.




• Using $a_1$ and $a_2$ that you found in the previous step, you should be able to compute $a$ and verify that it satisfy the condition.

Remark about your attempt:

• After you obtain $-20z+10az=10$, we have $z(2-a)=-1$. Now, we note that if $a=2$, then we get a contradiction, which is the value that you are looking for.




• Suppose $a ne 2$, then you can solve for $z$ in terms of $a$ and in turn solve for $x$ and then $y$.




• General writing remark for improving readability, you might like to label your equations and include how are each line obtain, which operation was performed.




• Things will be neater after you learn about Gaussian eliminations.

The planes do not intersect iff the system of equations has no solution. A necessary condition for that is that the determinant of the coefficient matrix vanishes. We have $$detbeginbmatrix1&-2&1\2&1&-3\1&1&-aendbmatrix = 10-5a.$$ There’s only one possible value of $a$ for which this determinant is equal to zero. Check this against the original system of equations to eliminate the possibility that you have an infinite number of solutions instead.