*412 what value(s) of h is b in plane spanned

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In summary, the value of $h$ for b to be in the plane spanned by $a_1$ and $a_2$ is 3. This can be found by setting up a system of linear equations and solving for $h$ using RREF. The solution is $h=3$, with $v=-7$ and $w=-2$.
  • #1
karush
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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..
 
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  • #2
karush said:
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
 
  • #3
topsquark said:
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...
 
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  • #4
karush said:
$\left[\begin{array}{r} 1\\3\\ -1 \end{array}\right]v+
\left[\begin{array}{r} -5\\-8\\2 \end{array}\right]w
=\left[\begin{array}{r} 3\\-5\\ \color{red}{h} \end{array}\right]$
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
 
  • #5
ok I see

however the OP only asked for h

mahalo
 
  • #6
karush said:
ok I see

however the OP only asked for h

mahalo
I know. It was just an FYI.

-Dan
 

Related to *412 what value(s) of h is b in plane spanned

1. What does it mean for a plane to be spanned by two vectors?

When a plane is spanned by two vectors, it means that any point on the plane can be reached by taking linear combinations of those two vectors. In other words, the two vectors are able to "span" or cover the entire plane.

2. How is the value of h related to the vectors b and the spanned plane?

The value of h represents the coefficient of one of the vectors in the linear combination used to reach a point on the spanned plane. It determines the position of the plane in relation to the origin and the direction in which it is spanned.

3. Can there be more than one value of h that satisfies the equation?

Yes, there can be multiple values of h that satisfy the equation. This is because there are an infinite number of possible linear combinations that can reach a point on the spanned plane.

4. How can the value(s) of h be determined?

The value(s) of h can be determined by solving the linear combination equation for the specific point on the spanned plane. This involves finding the coefficients that satisfy the equation and represent the position of the plane in relation to the origin.

5. What is the significance of finding the value(s) of h in this context?

Finding the value(s) of h allows us to understand the relationship between the vectors b and the spanned plane. It also helps us to visualize and manipulate the plane in mathematical calculations and applications.

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