Story One

Our idea here is to build a story about force and acceleration that is crazy insane. We hope that you laugh and that you see what the math can do.

We will use F=ma.

$m \overrightarrow{a} = \overrightarrow{F}$

We want calculations that solve for ‘a’ because acceleration is needed for calculations in Kinematics. If F is proportional to a, then a is proportional to F and we can write the following where c is a constant of proportiobality:

$cF = a$

For the above equation, c is a number so we call it a scalar. If we are limited to this equation, then we are doomed to a scenario where a and F will always be vectors pointing in the same direction differing only by magnitude. When making a, all we can do is scale F. Well, that’s actually how it works work celebration and force in physics, so this math is okay. But we said at the beginning of this paper that we wanted to go wild and crazy and insane. We will.

$m \cdot \overrightarrow{F} = \overrightarrow{a}$

Now m can be a matrix.

Using Algebra (see Proportionality) we can calculate that $c = \dfrac {1}{m}$ .

For the above to be true, the operation of ‘m’ “dotting” ‘a’ must produce a vector.

We call a mathematician on the phone, he tells us that a matrix operating on a vector will give us a vector.

Your Calculus III class taught you (or will teach you), that if we are doing the dot product with vectors the one on the left has to be a row and the one on the right has to be a column. We show this below:

$\begin{bmatrix} a_{n1} + a_{n2} + a_{n3} \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = a_{n1}b_1 + a_{n2}b_2 + a_{n3}b_3$

We can view the above equation as multiplying a 1×3 matrix by a 3×1 matrix.

Every time we matrix multiply a row vector by a column Vector, we get a scalar.

Now you have the basic ideas in mind. Let’s use them.

We want to solve for acceleration:

cF = a

To get there from F = ma we do the following:

$F = ma$

$\dfrac {1}{m}F = \dfrac{1}{m}ma$

$\dfrac {1}{m}F = a$

$c = \dfrac{1}{m}$

Assume the vectors are $\begin{bmatrix}F_x \\ F_y \\ F_z\end{bmatrix}$ and $\begin{bmatrix}a_x \\ a_y \\ a_z\end{bmatrix}$ :

$\begin{bmatrix} \dfrac{1}{m} & 0 & 0 \\ 0 & \dfrac{1}{m} & 0 \\ 0 & 0 & \dfrac{1}{m} \end{bmatrix} \begin{bmatrix} F_x \\ F_y \\ F_z \end{bmatrix} = \begin{bmatrix}a_x \\ a_y \\ a_z \end{bmatrix}$

$\dfrac{1}{m} F_x = a_x$
$\dfrac{1}{m} F_y = a_y$
$\dfrac{1}{m} F_z = a_z$

The above is what happens in reality as we know it. Let’s change something, just to see what happens:

$\begin{bmatrix} 10 \dfrac{1}{m} & 0 & 0 \\ 0 & \dfrac{1}{m} & 0 \\ 0 & 0 & \dfrac{1}{m} \end{bmatrix} \begin{bmatrix} F_x \\ F_y \\ F_z \end{bmatrix} = \begin{bmatrix}a_x \\ a_y \\ a_z \end{bmatrix}$

$10 \dfrac{1}{m} F_x = a_x$
$\dfrac{1}{m} F_y = a_y$
$\dfrac{1}{m} F_z = a_z$

Assume we have a constant force, but we can point in any direction. The more we point this force in the x direction, the greater the acceleration.

$\dfrac{1}{m} F_x = a_x$
$\dfrac{1}{m} F_y = a_y$
$\dfrac{1}{m} F_z = a_z$

Can you use this? Yes! If you are running from a constant velocity Predator, you want to run in the x-direction when you see the bad boy coming towards you.

Appendix A

$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{33}\\ a_{n1} & a_{n2} & a_{n3} \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix}$

If it is true that the dot symbol can be used to multiply aatrix against a first vector to make a second vector then we have the ability to change vector b to vector c, changing both magnitude and direction.

We could create a matrix that changes vector magnitude and rotates the vector by matrix multiplying a magnitude-changing matei

Important Disclaimer: all the discussions about changing magnitude and direction were strictly about the math. As a mathematician, you are expected to work behind the scenes in a way so that what the physicist sees is a zero change.

Physicists won’t know that you changed both (1) the magnitude of the components and (2) the length of the basis vectors in a way that the two changes cancelled out.

Appendix B

For our purposes, when we see a scalar it is usually some number like two or three. However, a scalar could be a complex number, so we want you to be aware of that.

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