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- Jan 26, 2012

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Thanks to Chris L T521 for submitting this problem!

Let $\mathbf{u}=u_1\mathbf{i}+u_2\mathbf{j}+u_3\mathbf{k}$ and $\mathbf{v}=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}$ be two vectors in $\mathbb{R}^3$. We define the

\[\|\mathbf{v}\| = \sqrt{v_1^2+v_2^2+v_3^2},\]

the

\[\mathbf{u}\cdot\mathbf{v}=u_1v_1 + u_2v_2 + u_3v_3,\]

and the

\[\mathbf{u}\times\mathbf{v} = \det\begin{pmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3\\ v_1 & v_2 & v_3\end{pmatrix}.\]

Express $\|\mathbf{u}\times\mathbf{v}\|^2 + (\mathbf{u}\cdot \mathbf{v})^2$ in terms of $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$ only.

No hints for this one either!

Remember to read the POTW submission guidelines to find out how to submit your answers!

Let $\mathbf{u}=u_1\mathbf{i}+u_2\mathbf{j}+u_3\mathbf{k}$ and $\mathbf{v}=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}$ be two vectors in $\mathbb{R}^3$. We define the

**norm**of the vector $\mathbf{v}$ by\[\|\mathbf{v}\| = \sqrt{v_1^2+v_2^2+v_3^2},\]

the

**dot product**of $\mathbf{u}$ and $\mathbf{v}$ by\[\mathbf{u}\cdot\mathbf{v}=u_1v_1 + u_2v_2 + u_3v_3,\]

and the

**cross product**of $\mathbf{u}$ and $\mathbf{v}$ by\[\mathbf{u}\times\mathbf{v} = \det\begin{pmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3\\ v_1 & v_2 & v_3\end{pmatrix}.\]

Express $\|\mathbf{u}\times\mathbf{v}\|^2 + (\mathbf{u}\cdot \mathbf{v})^2$ in terms of $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$ only.

No hints for this one either!

Remember to read the POTW submission guidelines to find out how to submit your answers!

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