Solving Vector Equations: Find Value of h

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SUMMARY

The discussion focuses on determining the value of h for which the vector b = (4, 1, h) lies in the plane spanned by the vectors a_1 = (1, 4, -2) and a_2 = (-2, -3, 7). The solution involves computing the determinant of the augmented matrix formed by these vectors. It is established that the vectors are coplanar when the determinant equals zero, leading to the conclusion that h must equal -17 for b to be in the plane defined by a_1 and a_2.

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Let [itex]a_1[/itex] = a column vector with 1, 4, -2; [itex]a_2[/itex] = a column vector with -2, -3, 7; and [itex]b[/itex] = 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 [itex]b[/itex] in the plane spanned by [itex]a_1[/itex] and [itex]a_2[/itex]?

I turned this into an augmented matrix but had trouble reducing it to RREF.
 
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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 [itex]C = a_1 \times a_2[/itex] as defining the plane spanned by [itex]a_1[/itex] and [itex]a_2[/itex]. Then h is in the plane if [itex]h \cdot C[/itex] 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:

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

Carl
 
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.

[tex]\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[/tex]
 

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