The torque, $\tau$, necessary to tighten a bolt is related to:
Meaning of the direction of $\myv \tau$? $\myv \tau$ is parallel to the axis of the bolt. $\myv \tau$ points in the direction that a standard, right-handed thread bolt would advance.
Which component of the force--parallel or perpendicular to $\myv r$--is more effective at tightening bolts?
We conclude that $$\tau = rF\sin\theta \nonumber$$ to motivate the...
If $\myv a$ and $\myv b$ are non-zero 3-d vectors, then the cross product $\myv a \times \myv b$ is: $$\myv a \times \myv b \equiv |\myv a| |\myv b| \sin \theta \uv n,$$ where $\theta$ is the smaller angle between $\myv a$ and $\myv b$ ($0\leq\theta\leq \pi$) and $\uv n$ is a unit vector perpendicular to $\myv a$ and $\myv b$ according to the right-hand rule: Curl r.h. fingers from $\myv a$ towards $\myv b$ and your thumb points in the same direction as $\uv n$.
With the right-hand rule, the order of the vectors in the cross product matters.
The length of the cross product, $|\myv a \times \myv b|$, is the area of the parallelogram determined by $\myv a$ and $\myv b$.
Grinding out the cross product from its components...
I remember how to compute the cross product by using the determinant of this 3$\times$3 matrix: $$\myv a \times \myv b = \text{det}\left| \begin{array}{ccc} \uv i & \uv j & \uv k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{array}\right|.$$
To compute the determinant,
add the red products,
and subtract
the blue products
$$\myv a \times \myv b=(a_2b_3-a_3b_2)\uv i + (a_3b_1-a_1b_3)\uv j+(a_1b_2-a_2b_1)\uv k.\nonumber$$
For example: $\myv a=\langle 1,-2,-4 \rangle$ and $\myv b=\langle 2,4,8\rangle$ $$\begineq \myv a \times \myv b &=& \det\left| \begin{array}{ccc} \uv i&\uv j&\uv k \\ 1 & -2 & -4 \\ 2 & 4 & 8 \end{array} \right|\\ &=&\\ &=&\\ &=& (-2*8 -(-4*4))\uv i+ (-4*2-(1*8))\uv j+ (1*4-(-2*2))\uv k\\ &=& -16\uv j +8\uv k=\langle 0, -16,8\rangle.\nonumber \endeq $$
There are many ways to form triple products. Of these, which are vector or scalar quantities?
The volume of the parallelipid determined by $\myv a$, $\myv b$, and $\myv c$ is: $$V=|\myv a \cdot (\myv b \times \myv c)|.$$
[Without proof] This may be calculated from: $$\myv a \cdot (\myv b \times \myv c) = \text{det}\left| \begin{array}{ccc} a_1&a_2&a_3\\ b_1&b_2&b_3\\ c_1&c_2&c_3 \end{array} \right|$$
$$\myv a \times (\myv b \times \myv c) =(\myv a \cdot \myv c)\myv b -(\myv a \cdot \myv b)\myv c.$$