FP3 - Volume of a tetrahedron using vectors

Hi,

Relative to a fixed origin O, the points A and B have position vectors

\mathbf{a}

metres and

\mathbf{b}

metres respectively, where:

\mathbf{a} = \begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix}

and

\mathbf{b} = \begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}

The point C moves such that the volume of the tetrahedron

OABC

is always

5 m^3

. Determine Cartesian equations for the locus of the point C.

I tried solving this as follows:

Volume

= \frac{1}{6} \begin{vmatrix} \begin{pmatrix} x \\ y \\ x \end{pmatrix} . \begin{pmatrix}\begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix} \times \begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix} \end{pmatrix} \end{vmatrix} = 5

\Rightarrow \begin{vmatrix} -6x + 15y - 9z \end{vmatrix} = 30

\Rightarrow -6x + 15y - 9z = 30

\Rightarrow 6x - 15y + 9z = 30

\Rightarrow 2x - 5y + 3z + 10 = 0

\Rightarrow 2x - 5y + 3z - 10 = 0

However my textbook only has the equation

2x - 5y + 3z + 10 = 0

I've tried to understand why they've ignored the modulus signs but I just can't see it. What am I missing?

Thank you :smile:

Original post by so it goes

What am I missing?

Nothing. The book's answer is incomplete.

Original post by ghostwalker

Nothing. The book's answer is incomplete.

Thank you

LaTex

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