Question Type 2: Rewriting vector operations Bootcamps

Show that for any vectors $\mathbf{u},\mathbf{v},\mathbf{w}$ in $\mathbb{R}^n$, the following holds: $|\mathbf{u}+\mathbf{v}+\mathbf{w}|^2 = |\mathbf{u}|^2+|\mathbf{v}|^2+|\mathbf{w}|^2 + 2(\mathbf{u}\cdot\mathbf{v} + \mathbf{v}\cdot\mathbf{w} + \mathbf{w}\cdot\mathbf{u}).$

Given two unit vectors $\mathbf{u}$ and $\mathbf{v}$ with $\mathbf{u}\cdot\mathbf{v}=\tfrac12$, compute $|\mathbf{u}+\mathbf{v}|^2$.

If $\mathbf{u}\cdot\mathbf{v}=2$, $|\mathbf{u}|=3$, and $|\mathbf{v}|=4$, compute $|\mathbf{u}-\mathbf{v}|$.

Vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $|\mathbf{u}|=1$, $|\mathbf{v}|=2$, and the angle between them is $60^\circ$. Compute $|\mathbf{u}\times\mathbf{v}|$.

Given nonzero vectors $\mathbf{u}$ and $\mathbf{v}$ with $\mathbf{u}\cdot\mathbf{v}=4$ and $|\mathbf{v}|=\sqrt{5}$, find the magnitude of the projection of $\mathbf{u}$ onto $\mathbf{v}$, i.e. $\bigl|\mathrm{proj}_{\mathbf{v}}\mathbf{u}\bigr|.$

Let $\mathbf{u}$ and $\mathbf{v}$ be vectors with $\mathbf{u}\cdot\mathbf{v}=-\tfrac13$, $|\mathbf{u}|=2$, and $|\mathbf{v}|=3$. Find $|\mathbf{u}-\mathbf{v}|^2$.

Let $\mathbf{u},\mathbf{v},\mathbf{w}$ be unit vectors with $\mathbf{u}\cdot\mathbf{v}=0$, $\mathbf{u}\cdot\mathbf{w}=\tfrac12$, and $\mathbf{v}\cdot\mathbf{w}=0$. Find $|\mathbf{u}+\mathbf{v}+\mathbf{w}|^2$.

In the plane, vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $|\mathbf{u}|=5$, $|\mathbf{v}|=7$, and $\mathbf{u}\cdot\mathbf{v}=14$. Find the area of the parallelogram determined by $\mathbf{u}$ and $\mathbf{v}$.

Show that for any vectors $\mathbf{u}$ and $\mathbf{v}$, the following identity holds: $|\mathbf{u}+\mathbf{v}|^2 + |\mathbf{u}-\mathbf{v}|^2 = 2\bigl(|\mathbf{u}|^2 + |\mathbf{v}|^2\bigr).$

For two unit vectors $\mathbf{u}$ and $\mathbf{v}$ with angle $\theta$ between them, express $|\mathbf{u}+\mathbf{v}|$ and $|\mathbf{u}-\mathbf{v}|$ in terms of $\theta$.

Prove the vector identity for any $\mathbf{u},\mathbf{v}\in\mathbb{R}^3$: $|\mathbf{u}\times\mathbf{v}|^2 = |\mathbf{u}|^2\,|\mathbf{v}|^2 - (\mathbf{u}\cdot\mathbf{v})^2.$

Let $\mathbf{u},\mathbf{v},\mathbf{w}$ satisfy $|\mathbf{u}|^2=|\mathbf{v}|^2=|\mathbf{w}|^2=4$, $\mathbf{u}\cdot\mathbf{v}=1$, $\mathbf{v}\cdot\mathbf{w}=2$, and $\mathbf{w}\cdot\mathbf{u}=3$. Compute $|\mathbf{u}+\mathbf{v}+\mathbf{w}|^2$.