va1={4,2,2,1};va2={-1,8,4,5};va3={-7,7,8,5}

vv1=va1

vv2=va2-((va2.vv1)/(vv1.vv1))vv1

vv3=va3-((va3.vv1)/(vv1.vv1))vv1-((va3.vv2)/(vv2.vv2))vv2

vv1.vv2

vv1.vv3

vv2.vv3

Norm[vv1]

Norm[vv2]

Norm[vv3]

Norm[vv1]Norm[vv2]Norm[vv3]

Transpose[{va1,va2,va3}]//MatrixForm

mA=Transpose[{va1,va2,va3}]

Det[Transpose[mA].mA]

With the corrected vector a3:

va1={4,2,2,1};va2={-1,8,4,5};va3c={-7,7,-8,5}

vv1=va1

vv2=va2-((va2.vv1)/(vv1.vv1))vv1

vv3c=va3c-((va3c.vv1)/(vv1.vv1))vv1-((va3c.vv2)/(vv2.vv2))vv2

vv1.vv2

vv1.vv3c

vv2.vv3c

Norm[vv1]

Norm[vv2]

Norm[vv3c]

Norm[vv1]Norm[vv2]Norm[vv3c]

Transpose[{va1,va2,va3c}]//MatrixForm

mAc=Transpose[{va1,va2,va3c}]

(Sqrt[Det[Transpose[mAc].mAc]])

To use the code on my website, I use

TeXForm[mAc]

And of course

Norm[vv1]Norm[vv2]Norm[vv3c]==(Sqrt[Det[Transpose[mAc].mAc]])