Solving Vector Equations: Find Value of h

[tandoorichicken]

Let $a_1$ = a column vector with 1, 4, -2; $a_2$ = a column vector with -2, -3, 7; and $b$ = a column vector with entries 4, 1, h.
(I hope this is an adequate description. I forgot how to write pretty matrices in tex ^_^;)

For what values of h is $b$ in the plane spanned by $a_1$ and $a_2$?

I turned this into an augmented matrix but had trouble reducing it to RREF.

 

[CarlB]

I had the best success in vectors when I put everything in terms of dot products and cross products. In this problem, one can take $C = a_1 \times a_2$ as defining the plane spanned by $a_1$ and $a_2$. Then h is in the plane if $h \cdot C$ is zero. That is, if C is perpendicular to h.

Click on this example to be reminded how to format matrices in LaTex with various boundary definitions &c:

$$\left( \left[ \begin{array}{ccc}<br /> 0 & 1 & 2 \\<br /> 3 & 4 & 5 \end{array} \right| \right)$$

Carl

 

[TD]

Put them in a matrix and compute the determinant. If det is 0, then the vector are coplanar and thus, every vector is in the plane span by the other two.

$$\left| {\begin{array}{*{20}c}<br /> 1 & { - 2} & 4 \\<br /> 4 & { - 3} & 1 \\<br /> { - 2} & 7 & h \\<br /> <br /> \end{array} } \right| = 0 \Leftrightarrow h = - 17$$