Vector and matrix toolbox for spatial models

Before using tensors or a curvilinear vector formula, we need the arithmetic of vectors, the action of a matrix and the geometry of determinants. The same toolbox describes observation systems, forces, oriented areas and coordinate frames. We build it explicitly within this course rather than relying on an unstated linear-algebra chapter elsewhere.

1 Vectors dot products and projections

Vectors in $\mathbb R^2$ or $\mathbb R^3$ add coordinatewise and scale by real numbers. The dot product $u\cdot v=\sum u_iv_i$ gives Euclidean length $\|u\|=\sqrt{u\cdot u}$. Expanding $\|u-tv\|^2\ge0$ and minimizing in $t$ proves Cauchy–Schwarz, $|u\cdot v|\le\|u\|\|v\|$. Equality means linear dependence, including zero-vector cases.

For a unit direction $n$, $u\cdot n$ is the signed component along it, with vector component $(u\cdot n)n$. For nonzero $n$ not necessarily unit, the projection is $[(u\cdot n)/(n\cdot n)]n$. For $u=(3,1,2)$ and $n=(1,0,1)$, the projection is $(5/2,0,5/2)$; the residual $(1/2,1,-1/2)$ has zero dot product with $n$.

2 Cross products and determinants

In three dimensions,

$$u\times v=(u_2v_3-u_3v_2,\ u_3v_1-u_1v_3,\ u_1v_2-u_2v_1).$$

Direct substitution shows it is perpendicular to both inputs. Expanding its squared norm gives $\|u\times v\|^2=\|u\|^2\|v\|^2-(u\cdot v)^2$, so its magnitude is the parallelogram area. Its sign follows the right-hand rule, and reversing the input order reverses the vector.

For a $2\times2$ matrix, determinant is $a_{11}a_{22}-a_{12}a_{21}$. For a $3\times3$ matrix with columns $u,v,w$, define $\det A=u\cdot(v\times w)$, giving signed volume. Expansion along the first row is

$$a_{11}(a_{22}a_{33}-a_{23}a_{32}) -a_{12}(a_{21}a_{33}-a_{23}a_{31}) +a_{13}(a_{21}a_{32}-a_{22}a_{31}).$$

The determinant is linear in each column separately and changes sign under a column exchange, as the triple product shows. The same row properties follow from the expansion. Row addition preserves it, a row scaling scales it, and a row swap negates it. Elimination therefore proves a square matrix is invertible exactly when its determinant is nonzero. A zero signed volume means the three column directions fail to span spatial volume.

3 Matrix action and Gaussian elimination

For $A\in\mathbb R^{m\times n}$, $(Ax)_i=\sum_j a_{ij}x_j$. This is a linear map: $A(su+tv)=sAu+tAv$. Its columns are the images of standard coordinate vectors. Composition is multiplication, $(AB)_{ij}=\sum_k a_{ik}b_{kj}$, with compatible dimensions.

To solve $Ax=b$, perform reversible row operations on the full augmented matrix. A zero row with nonzero right-hand side is inconsistent; free variables give nonuniqueness. Row operations on $A$ alone while ignoring $b$ change the problem.

Worked observation system. Suppose three hypothetical scaled observation channels give

$$u+v+w=6,\quad u+2v+3w=14,\quad2u+v+w=7.$$

Subtract the first equation from the third to get $u=1$. Then $v+w=5$ and $2v+3w=13$, yielding $w=3$, $v=2$. All original observations verify the result. The coefficient determinant is 1, confirming uniqueness. Physical observations need a separately justified linear model and compatible units; solving the equations does not validate that model.

4 Orthogonal matrices and the role of orientation

A real square matrix is orthogonal when $Q^TQ=I$, equivalently its columns are orthonormal. Then $\|Qv\|^2=v^TQ^TQv=\|v\|^2$. Independent square columns make $Q$ invertible, and $Q^{-1}=Q^T$. Determinant multiplication gives $(\det Q)^2=1$, so it is $+1$ or $-1$.

For a rotation around the $z$ axis,

$$Q=\begin{pmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}.$$

Its determinant is 1. The diagonal matrix $\operatorname{diag}(-1,1,1)$ also preserves lengths but has determinant $-1$, reversing orientation. Thus not every orthogonal transformation is a proper rotation.

Distinguish a physical transformation from a coordinate change. Applying $Q$ physically to the old vector $(1,0,0)$ gives $(\cos\theta,\sin\theta,0)$ in old coordinates. If instead the new basis vectors are the columns of $Q$, the unchanged physical vector has new coordinates $Q^T(1,0,0)=(\cos\theta,-\sin\theta,0)$. The opposite sign is not a contradiction; the questions differ. This distinction underlies the tensor component laws and the varying coordinate frames developed later.

Zero cases and the determinant multiplication law

In the Cauchy–Schwarz argument, handle $v=0$ first, when equality holds. Otherwise substitute $t=(u\cdot v)/\|v\|^2$ into $\|u-tv\|^2\ge0$, obtaining $\|u\|^2-(u\cdot v)^2/\|v\|^2\ge0$. Multiplication by the positive denominator and taking square roots gives the inequality. In this nonzero-$v$ case, equality means $u=tv$; together with the zero-vector case this is exactly linear dependence.

For the two- and three-dimensional determinant formulas in this toolbox, alternation and columnwise linearity prove multiplication. Fix $A$ and regard $D(b_1,\ldots,b_n)=\det(Ab_1,\ldots,Ab_n)$ as a function of the columns of $B$, where $n=2$ or $3$. Expand each column in the standard basis. Terms with repeated basis vectors vanish by alternation; each remaining term is its permutation sign times $D(e_1,\ldots,e_n)=\det A$. The sum of the corresponding coefficient products is precisely the two- or three-dimensional expansion for $\det B$. Thus $\det(AB)=\det A\det B$. The displayed expansions also give $\det(A^T)=\det A$ by exchanging row and column indices. In particular $Q^TQ=I$ gives $(\det Q)^2=1$. This proves the determinant facts used for the spatial coordinate and tensor examples in this toolbox.

5 Graded practice

E1 Foundation. Compute $(1,0,1)\times(0,2,0)$ and its magnitude. Interpret the magnitude geometrically.

E2 Determinant. Calculate $\det\begin{pmatrix}1&2&0\\0&1&3\\0&0&2\end{pmatrix}$ and decide invertibility.

E3 Orthogonality. Verify $Q=\operatorname{diag}(-1,1,1)$ is orthogonal. Is it a proper rotation? What happens to signed triple products?

E4 Elimination. Solve $2u+v=4$, $v+w=5$, $u+w=4$ and verify all equations.

6 Full solutions

S1. The cross product is $(-2,0,2)$, of magnitude $2\sqrt2$. It is the area of the parallelogram spanned by the two input vectors; a triangle with those sides has half that area.

S2. Triangularity or direct expansion gives $1\cdot1\cdot2=2$. It is nonzero, so the matrix is invertible and its three column vectors are independent.

S3. $Q^TQ=I$, but determinant is $-1$, so it is a reflection rather than a proper rotation. It multiplies every oriented triple product by $-1$ while preserving absolute volumes and lengths.

S4. From $u+w=4$, $w=4-u$; then $v=1+u$. The first equation gives $3u+1=4$, so $u=1$, $v=2$, $w=3$. Checks are $2+2=4$, $2+3=5$, and $1+3=4$.

Summary

Dot and cross products encode spatial lengths, projections, areas and orientations. Determinants detect invertibility and signed volume. Gaussian elimination solves a stated linear observation model, while orthogonal matrices distinguish length preservation from orientation preservation. Physical rotations and coordinate changes use related matrices but answer different questions.