vector equations

**tandoorichicken** · Oct3-05, 09:59 PM

Let

= a column vector with 1, 4, -2;

= a column vector with -2, -3, 7; and

= 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

in the plane spanned by

and

?

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

**CarlB** · Oct4-05, 02:02 AM

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 $LaTeX Code: C = a_1 \\times a_2$ as defining the plane spanned by

and

. Then h is in the plane if $LaTeX Code: h \\cdot C$ 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:

$LaTeX Code: \\left( \\left[ \\begin{array}{ccc} 0 & 1 & 2 \\\\ 3 & 4 & 5 \\end{array} \\right| \\right)$

Carl

TD · Oct4-05, 06:33 AM

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.

$LaTeX Code: \\left| {\\begin{array}{*{20}c} 1 & { - 2} & 4 \\\\ 4 & { - 3} & 1 \\\\ { - 2} & 7 & h \\\\ \\end{array} } \\right| = 0 \\Leftrightarrow h = - 17$