IB Mathematics Analysis and Approaches Question Type 3: Simplifying Vector Operations with Dot and Cross Product Exercises

Simplify the dot product expression $(\mathbf{v}+\mathbf{w})\cdot (\mathbf{v}-\mathbf{w})$ in terms of $|\mathbf{v}|$ and $|\mathbf{w}|$.

Let $\mathbf{m}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$ and $\mathbf{n}=\left(\begin{array}{c}2\\ 1\\ 3\end{array}\right)$. Calculate $(\mathbf{m}+2\mathbf{n})\cdot (3\mathbf{m}-\mathbf{n})$.

Compute the cross product $(\mathbf{i}+2\mathbf{j})\times (3\mathbf{i}-\mathbf{j}+\mathbf{k})$.

Let $\mathbf{u}$ and $\mathbf{v}$ be perpendicular vectors with $|\mathbf{u}|=2$ and $|\mathbf{v}|=3$. If $\mathbf{w}=\mathbf{u}-2\mathbf{v}$, find $|\mathbf{w}|$.

Given $\mathbf{v}\times \mathbf{w}=\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}2\\ 2\\ 2\end{array}\right)$, find $\mathbf{w}\times \left(\mathbf{v}+\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)\right)$.

Given the scalar triple product $\mathbf{v}\cdot (\mathbf{w}\times \mathbf{u})=5$, determine $\mathbf{w}\cdot (\mathbf{u}\times \mathbf{v})$.

Given $\mathbf{a}\times (\mathbf{b}+\mathbf{c})=(1,2,3)$ and $\mathbf{a}\times \mathbf{b}=(0,1,-1)$, find $\mathbf{a}\times \mathbf{c}$.

Given vectors $\mathbf{a}$ and $\mathbf{b}$ with $\mathbf{a}\cdot \mathbf{b}=5$, $|\mathbf{a}|=3$, $|\mathbf{b}|=4$, compute $\mathbf{a}\cdot (2\mathbf{a}-\mathbf{b})$.

Suppose $\mathbf{v}\times \mathbf{w}=(3,1,-2)$, $\mathbf{w}\times \mathbf{u}=(0,2,1)$ and $\mathbf{u}\times \mathbf{v}=(-3,1,1)$. Compute $(\mathbf{v}+\mathbf{w}+\mathbf{u})\times (\mathbf{w}+\mathbf{u})$.

Verify that $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{w})=0$ for $\mathbf{v}=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}4\\ 5\\ 6\end{array}\right)$ by direct computation.

Given $\mathbf{u}\times \mathbf{v}=(3,-1,2)$ and $\mathbf{v}\times \mathbf{w}=(1,4,-3)$, compute $(\mathbf{u}+\mathbf{w})\times \mathbf{v}$.