# [SOLVED]211 For what value(s) of h is b in the plane spanned

#### karush

##### Well-known member
For what value(s) of $h$ is b in the plane spanned by $a_1$ and $a_2$
$$a_1=\left[\begin{array}{r} 1\\3\\ -1 \end{array}\right], a_2=\left[\begin{array}{r} -5\\-8\\2 \end{array}\right], b =\left[\begin{array}{r} 3\\-5\\ \color{red}{h} \end{array}\right]$$

ok this should be obvious but I don't see it..

#### topsquark

##### Well-known member
MHB Math Helper
For what value(s) of $h$ is b in the plane spanned by $a_1$ and $a_2$
$$a_1=\left[\begin{array}{r} 1\\3\\ -1 \end{array}\right], a_2=\left[\begin{array}{r} -5\\-8\\2 \end{array}\right], b =\left[\begin{array}{r} 3\\-5\\ \color{red}{h} \end{array}\right]$$

ok this should be obvious but I don't see it..
Hint: If b is in the plane formed by a_1 and a_2 then it has to be a linear combination of a_1 and a_2. ie. $$\displaystyle b = v a_1 + w a_2$$ for some v, w constants.

Can you finish?

-Dan

#### karush

##### Well-known member
Hint: If b is in the plane formed by a_1 and a_2 then it has to be a linear combination of a_1 and a_2. ie. $$\displaystyle b = v a_1 + w a_2$$ for some v, w constants.

Can you finish?

-Dan
$\left[\begin{array}{r} 1\\3\\ -1 \end{array}\right]v+ \left[\begin{array}{r} -5\\-8\\2 \end{array}\right]w =\left[\begin{array}{r} 3\\-5\\ \color{red}{h} \end{array}\right]$
so then the augmented matrix would be
$\left[\begin{array}{rr|r}1 & -5 & 3 \\ 3 & -8 & -5 \\ -1 & 2 & h \end{array}\right]$
then RREF
$\left[ \begin{array}{cc|c} 1 & 0 & -7 \\0 & 1 & -2 \\ 0 & 0 & h - 3 \end{array} \right]$
so $h=3$ following would be $v=7$ and $w=2$

hopefully....

Last edited:

#### topsquark

##### Well-known member
MHB Math Helper
$\left[\begin{array}{r} 1\\3\\ -1 \end{array}\right]v+ \left[\begin{array}{r} -5\\-8\\2 \end{array}\right]w =\left[\begin{array}{r} 3\\-5\\ \color{red}{h} \end{array}\right]$
so then the augmented matrix would be
$\left[\begin{array}{rr|r}1 & -5 & 3 \\ 3 & -8 & -5 \\ -1 & 2 & h \end{array}\right]$
then RREF
$\left[ \begin{array}{cc|c} 1 & 0 & -7 \\0 & 1 & -2 \\ 0 & 0 & h - 3 \end{array} \right]$
so $h=3$ following would be $v=7$ and $w=2$

hopefully....
I didn't go looking for it but somehow you are off by a sign. v = -7 and w = -2 and h = 3 is the solution.

-Dan

#### karush

##### Well-known member
ok I see

however the OP only asked for h

mahalo

#### topsquark

##### Well-known member
MHB Math Helper
ok I see

however the OP only asked for h

mahalo
I know. It was just an FYI.

-Dan