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 $\mathbf{m} = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$ and $\mathbf{n} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}$. 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 $\boldsymbol{v} \times \boldsymbol{w} = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$ and $\boldsymbol{w} = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$, find $\boldsymbol{w} \times \left(\boldsymbol{v} + \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\right)$.

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 $\mathbf{v} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}$ by direct computation.

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