Find the scalar triple product u*(v imes w). Show work. u=(-3,1,3),v=(5,1,-2),w=(-2,1,5). Find the scalar triple product u : (v x w). Show work u=(-3,1,3),v=(5,1,-2),w=(-2,1,5)
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To do this, we will use the formula for the cross product of two 3-dimensional vectors: v x w = (v2w3 - v3w2, v3w1 - v1w3, v1w2 - v2w1) Show more…
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Calculate the scalar triple product u · (v × w), where u = (1, 1, 0), v = (3, -2, 2), and w = (4, -1, 2).
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Scalar triple product Another operation with vectors is the scalar triple product, defined to be $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}),$ for vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ in $\mathbb{R}^{3}$ Express $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ in terms of their components and show that $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})$ equals the determinant $\left|\begin{array}{lll}u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3}\end{array}\right|$
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Proof Prove that the triple scalar product of $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ is given by $$ \mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=\left|\begin{array}{lll}{u_{1}} & {u_{2}} & {u_{3}} \\ {v_{1}} & {v_{2}} & {v_{3}} \\ {w_{1}} & {w_{2}} & {w_{3}}\end{array}\right| $$
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