IB Mathematics Analysis and Approaches Question Type 2: Rewriting Vector Operations Exercises

Let $\mathbf{u},\mathbf{v},\mathbf{w}$ be unit vectors with $\mathbf{u}\cdot \mathbf{v}=0$, $\mathbf{u}\cdot \mathbf{w}=\frac{1}{2}$, 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 $\mathbf{v}$.

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 $|\mathbf{u}\times \mathbf{v}|$.

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 $\mathbf{u}$ and $\mathbf{v}$ be vectors with $\mathbf{u}\cdot \mathbf{v}=-\frac{1}{3}$, $|\mathbf{u}|=2$, and $|\mathbf{v}|=3$. Find $|\mathbf{u}-\mathbf{v}{|}^2$.

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 $\theta$.

Given two unit vectors $\mathbf{u}$ and $\mathbf{v}$ with $\mathbf{u}\cdot \mathbf{v}=\frac{1}{2}$, compute $|\mathbf{u}+\mathbf{v}{|}^2$.

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

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

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 $\mathbf{w}\cdot \mathbf{u}=3$. Find $|\mathbf{u}+\mathbf{v}+\mathbf{w}{|}^2$.

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. $$\left|{\mathrm{proj}}_{\mathbf{v}}\mathbf{u}\right|.$$