Question Type 3: Simplifying vector operations with dot and cross product Bootcamps

Verify that $v\cdot(v\times w)=0$ for $v=(1,2,3)$ and $w=(4,5,6)$ by direct computation.

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

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

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

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

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

Let $m = (1,0,-1)$ and $n = (2,1,3)$. Calculate $(m + 2n)\cdot(3m - n)$.

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

Given $v \times w = -(1,0,1)$ and $w = (2,2,2)$, find $w \times \bigl(v + (1,0,1)\bigr)$.

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

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