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# Use the scalar triple product to verify that the vectors $u = i + 5j - 2k$, $v = 3i - j$, and $w = 5i + 9j - 4k$ are coplanar.

## \begin{aligned} \mathbf{v} \cdot(\mathbf{u} \times \mathbf{w}) &=<3 .-1.0>\cdot(<1,5,-2>x<5,9,-4>) \\ &=<3,-1,0>\cdot(<-2,-6,-16>) \\ &=-6+6+0 \\ &=0 \end{aligned}

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Okay, there's a theorem and it says okay if the triple product okay equals zero the victors our complaint to. So all we have to do is get the triple product to equal to zero. And the triple product just means The Determinant 1 5 -2. 3 -10 5. -4. Okay so I'm not sure how you know how to do it, but here's one way you copy the first column, you copy the second column, Then you go diagonally. So 1 -1 -4. That's four plus five. plus -239 -6 times 9 -6 times nine. Yeah. Oh sorry minus 50 for minus now. Go the other way five minus 12. So positive 10 plus zero plus mm 15 times minus four minus 60. And so we get minus 50 for minus 50 minus minus 50 which equals zero. So completely.

Oklahoma State University

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