Solving vector equations in Mathematica

Question

I'm trying to figure out how to use Mathematica to solve systems of equations where some of the variables and coefficients are vectors. A simple example would be something like where I know A, V, and the magnitude of P, and I have to solve for t and the direction of P. (Basically, given two rays A and B, where I know everything about A but only the origin and magnitude of B, figure out what the direction of B must be such that it intersects A.) Now, I know how to solve this sort of thing by hand, but that's slow and error-prone, so I was hoping I could use Mathematica to speed things along and error-check me. However, I can't see how to get Mathematica to symbolically solve equations involving vectors like this. I've looked in the VectorAnalysis package, without finding anything there that seems relevant; meanwhile the Linear Algebra package only seems to have a solver for linear systems (which this isn't, since I don't know t or P, just |P|). I tried doing the simpleminded thing: exp…

Accepted Answer

With Mathematica 7.0.1.0

```
Clear[A, V, P];
A = {1, 2, 3};
V = {4, 5, 6};
P = {P1, P2, P3};
Solve[A + V t == P, P]
```

outputs:

```
{{P1 -> 1 + 4 t, P2 -> 2 + 5 t, P3 -> 3 (1 + 2 t)}}
```

Typing out P = {P1, P2, P3} can be annoying if the array or matrix is large.

```
Clear[A, V, PP, P];
A = {1, 2, 3};
V = {4, 5, 6};
PP = Array[P, 3];
Solve[A + V t == PP, PP]
```

outputs:

```
{{P[1] -> 1 + 4 t, P[2] -> 2 + 5 t, P[3] -> 3 (1 + 2 t)}}
```

Matrix vector inner product:

```
Clear[A, xx, bb];
A = {{1, 5}, {6, 7}};
xx = Array[x, 2];
bb = Array[b, 2];
Solve[A.xx == bb, xx]
```

outputs:

```
{{x[1] -> 1/23 (-7 b[1] + 5 b[2]), x[2] -> 1/23 (6 b[1] - b[2])}}
```

Matrix multiplication:

```
Clear[A, BB, d];
A = {{1, 5}, {6, 7}};
BB = Array[B, {2, 2}];
d = {{6, 7}, {8, 9}};
Solve[A.BB == d]
```

outputs:

```
{{B[1, 1] -> -(2/23), B[2, 1] -> 28/23, B[1, 2] -> -(4/23), B[2, 2] -> 33/23}}
```

The dot product has an infix notation built in just use a period for the dot.

I do not think the cross product does however. This is how you use the Notation package to make one. "X" will become our infix form of Cross. I suggest coping the example from the Notation, Symbolize and InfixNotation tutorial. Also use the Notation Palette which helps abstract away some of the Box syntax.

```
Clear[X]
Needs["Notation`"]
Notation[x_ X y_\[DoubleLongLeftRightArrow]Cross[x_, y_]]
Notation[NotationTemplateTag[
  RowBox[{x_,  , X,  , y_,  }]] \[DoubleLongLeftRightArrow] 
  NotationTemplateTag[RowBox[{ , 
RowBox[{Cross, [, 
RowBox[{x_, ,, y_}], ]}]}]]]
{a, b, c} X {x, y, z}
```

outputs:

```
{-c y + b z, c x - a z, -b x + a y}
```

The above looks horrible but when using the Notation Palette it looks like:

```
Clear[X]
Needs["Notation`"]
Notation[x_ X y_\[DoubleLongLeftRightArrow]Cross[x_, y_]]
{a, b, c} X {x, y, z}
```

I have run into some quirks using the notation package in the past versions of mathematica so be careful.

Solving vector equations in Mathematica

Solution

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