Question

Let a1equals[Start 3 By 1 Matrix 1st Row 1st Column 1 2nd Row 1st Column 2 3rd Row 1st Column negative 1 EndMatrix ]​, a2equals[Start 3 By 1 Matrix 1st Row 1st Column negative 7 2nd Row 1st Column negative 7 3rd Row 1st Column 2 EndMatrix ]​, and bequals[Start 3 By 1 Matrix 1st Row 1st Column 3 2nd Row 1st Column negative 22 3rd Row 1st Column h EndMatrix ]. For what​ value(s) of h is b in the plane spanned by a1 and a2​?

Answer 1

Answer:

For h= 25,  b in the plane spanned by a1 and a2​

Step-by-step explanation:

$a1= \left[\begin{array}{c}1\\2\\-1\end{array}\right] \\a2 = \left[\begin{array}{c}-7\\-7\\2\end{array}\right] \\\\b = \left[\begin{array}{c}3\\-22\\h\end{array}\right]$

we have to find value of h for which  b in the plane spanned by a1 and a2.

For this the linear systems given by the  following augmented matrix must be consistent.

$\left[\begin{array}{cc|c}1&-7&3\\2&-7&-22\\-1&2&h\end{array}\right]$

Reduce the augmented matrix into row echelon form:

$R_{2} - 2R_{1} , R_{3} + R_{1}\\\\\left[\begin{array}{cc|c}1&-7&3\\0&7&-28\\-0&-5&h+3\end{array}\right]\\\\7R_{3}+5R_{2}\\\\\left[\begin{array}{cc|c}1&-7&3\\0&7&-28\\-0&0&7h-175\end{array}\right]$

For system to be consistent:

                         $7h-175 =0\\7h=175\\h=25$