vector equations

[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}
0 & 1 & 2 \\
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}
1 & { - 2} & 4 \\
4 & { - 3} & 1 \\
{ - 2} & 7 & h \\

\end{array} } \right| = 0 \Leftrightarrow h = - 17