Linear Algebra – Dot & Cross Product Lecture Notes

Administrative & Logistical Notes

• All material is provided under Section 113P of the Australian Copyright Act 1968 – do not redistribute.

• Maths Drop-in Centre: Mon–Fri, Weeks 1-12, 2-4 pm, ground floor of the Mathematical Sciences Institute building.

  • Bring any mathematics question; staff are waiting to help.

• Slides policy

  • Un-annotated slides go up before lecture; annotated slides (cleaned-up for typos/drawing quality) uploaded afterwards.

Recap: Vector Addition & Scaling

• Previously learnt how to add vectors and multiply by scalars.

• Today: two ways to multiply vectors:

  1. Dot product ($\mathbf a\cdot\mathbf b$) – produces a scalar.

  2. Cross product ($\mathbf a\times\mathbf b$) – produces a vector (and works only in $\mathbb R^3$).

Dot Product (Stewart 10.3)

Definition

• For $\mathbf u=(u<em>1,u</em>2,u<em>3),\;\mathbf v=(v</em>1,v<em>2,v</em>3)\in \mathbb R^3$
  $\mathbf u\cdot\mathbf v \,=\, u<em>1v</em>1+u<em>2v</em>2+u<em>3v</em>3.$

• Extends to any equal dimension $n$ by summing component-wise products.

• Output is a scalar (number).

Basic Algebraic Properties (usable facts)

1. $\mathbf u\cdot\mathbf v = \mathbf v\cdot\mathbf u$ (commutative).

2. $\mathbf u\cdot(\mathbf v+\mathbf w)=\mathbf u\cdot\mathbf v+\mathbf u\cdot\mathbf w$ (distributive).

3. $(c\mathbf u)\cdot\mathbf v = c\,(\mathbf u\cdot\mathbf v)$ for any scalar $c$.

4. $\mathbf u\cdot\mathbf u = |\mathbf u|^{2}.$

5. $|\mathbf u|=\sqrt{\mathbf u\cdot\mathbf u}$ – links dot product with length.

Geometric Interpretation: Angle Formula

• If $\theta$ is the smaller angle ($0\le\theta\le\pi$) between non-zero $\mathbf a,\mathbf b$ then
  $\mathbf a\cdot\mathbf b = |\mathbf a|\,|\mathbf b|\,\cos\theta.$
  Equivalently,
  $\cos\theta = \frac{\mathbf a\cdot\mathbf b}{|\mathbf a|\,|\mathbf b|}.$

• Because $\cos\theta$ is injective on $[0,\pi]$, the ratio determines a unique $\theta$.

Cauchy–Schwarz (special case here)

• $-1\le \frac{\mathbf a\cdot\mathbf b}{|\mathbf a|\,|\mathbf b|}\le 1.$
  Ensures the above formula always legitimate.

Signs & Orthogonality

Using the cosine graph on $[0,\pi]$:

• \mathbf a\cdot\mathbf b>0 ⟺ 0<\theta<\tfrac\pi2 (acute).

• $\mathbf a\cdot\mathbf b=0$ ⟺ $\theta=\tfrac\pi2$ or one vector is $\mathbf0$.

  • Definition: vectors are orthogonal / perpendicular ⇔ their dot product is zero.

• \mathbf a\cdot\mathbf b<0 ⟺ \tfrac\pi2<\theta<\pi (obtuse).

Worked Examples

• Example 1: $\begin{aligned}\mathbf u&amp;=(1,2,3)\ \mathbf v&amp;=(-2,4,1)\end{aligned}$
  $\mathbf u\cdot\mathbf v = 1(-2)+2(4)+3(1)=9.$

• Example 2 (orthogonality test): $\mathbf a=(2,2,-1),\;\mathbf b=(5,-4,2).$
  $\mathbf a\cdot\mathbf b=2\cdot5+2(-4)+(-1)\cdot2=10-8-2=0\;\Rightarrow\; \mathbf a\perp\mathbf b.$

Projection Concepts

Scalar (signed) Projection

• Component of $\mathbf b$ along $\mathbf a$: $\operatorname{comp}_{\mathbf a}\mathbf b = \frac{\mathbf a\cdot\mathbf b}{|\mathbf a|}.$

  • Positive if pointing roughly same way as $\mathbf a$, negative if opposite.

Vector Projection

• Actual shadow vector on the line spanned by $\mathbf a$: $\operatorname{proj}_{\mathbf a}\mathbf b = \left(\frac{\mathbf a\cdot\mathbf b}{|\mathbf a|^{2}}\right)\mathbf a.$

  • Obtained by multiplying scalar projection by the unit vector $\dfrac{\mathbf a}{|\mathbf a|}$.

Quick geometric cases

• Parallel, same direction → projection equals $\mathbf b$.

• Parallel, opposite direction → projection equals $-\mathbf b$.

• Orthogonal → projection is zero vector.

Interpretations & Uses

• $\mathbf a\cdot\mathbf b$ measures how much $\mathbf b$ points along $\mathbf a$ (or vice-versa, symmetry).

• In physics: determines work done, component of a force along a direction, etc.

• Computation-friendly: only adds/multiplies; ideal for programming.

Cross Product (Stewart 10.4)

Definition (works only in $\mathbb R^3$)

For $\mathbf a=(a<em>1,a</em>2,a<em>3),\;\mathbf b=(b</em>1,b<em>2,b</em>3)$,
$\mathbf a\times\mathbf b = \bigl(a<em>2 b</em>3 - a<em>3 b</em>2,\; a<em>3 b</em>1 - a<em>1 b</em>3,\; a<em>1 b</em>2 - a<em>2 b</em>1\bigr).$

• Output is a vector.

• Also defined in $\mathbb R^7$ (advanced topic), not in other dimensions.

Determinant Mnemonic (cofactor expansion)

\mathbf a\times\mathbf b = \begin{vmatrix}
\mathbf\hat i & \mathbf\hat j & \mathbf\hat k\
a1 & a2 & a3\ b1 & b2 & b3\end{vmatrix}
(Caveat: matrix contains basis vectors – purely a memory aid, not rigorous matrix algebra.)

Key Properties

• Orthogonality: $\mathbf a\times\mathbf b\perp\mathbf a$ and $\mathbf a\times\mathbf b\perp\mathbf b.$

• Anti-commutative: $\mathbf a\times\mathbf b = -\,\mathbf b\times\mathbf a.$

• Distributive: $\mathbf a\times(\mathbf b+\mathbf c)=\mathbf a\times\mathbf b+\mathbf a\times\mathbf c.$

• Scalar interaction: $(c\mathbf a)\times\mathbf b = c(\mathbf a\times\mathbf b).$

Length & Angle Relationship

• If $\theta$ is the smaller angle between non-zero $\mathbf a,\mathbf b$:
  $|\mathbf a\times\mathbf b| = |\mathbf a|\,|\mathbf b|\,\sin\theta.$
  Hence
  $\sin\theta = \frac{|\mathbf a\times\mathbf b|}{|\mathbf a|\,|\mathbf b|}.$

• Consequences

  • Cross‐product magnitude is maximal when $\theta=\tfrac\pi2$ (vectors orthogonal).

  • Zero when vectors are parallel or anti-parallel.

Right-Hand Rule (orientation)

• Point fingers along $\mathbf a$, curl towards $\mathbf b$; thumb indicates $\mathbf a\times\mathbf b.$

• Reversing order reverses direction.

Area Interpretation

• Parallelogram spanned by $\mathbf a,\mathbf b$ has area
  $\text{Area}=|\mathbf a\times\mathbf b|.$

• Useful for computing surface normals, torque, magnetic force, etc.

Example Sketch (exercise)

Let $\mathbf a=(1,0,2),\;\mathbf b=(3,1,-1).$ Compute using determinant mnemonic.

Dot vs Cross – Comparative Table

• Dot → scalar, cos angle, measures alignment.

• Cross → vector, sin angle, produces perpendicular direction & area.

• Dot zero ⇒ orthogonal; Cross magnitude maximal ⇒ orthogonal.

• Dot commutative; Cross anti-commutative.

Additional Observations & Star Tasks ★

• ★ Geometric proofs: verify each dot-product property using the parallelogram law & cosine rule.

• ★ Planes in $\mathbb R^3$: show two non-collinear vectors define one plane; collinear vectors define infinitely many – connect to angle discussion.

• ★ Why swapping $\mathbf a,\mathbf b$ in projection formulas still holds (dot symmetry).

• ★ Research: why only $\mathbb R^3$ and $\mathbb R^7$ admit a binary vector product with similar properties.

Practical Tips for Assignments & MATLAB Grader

• Enter exact symbolic answers (e.g. $\arccos\bigl(2/(3\sqrt{38})\bigr)$) – avoid intermediate decimal rounding.

• Remember $\operatorname{comp}_{\mathbf a}\mathbf b$ may be negative; indicate sign explicitly.

• Always check “zero” answers: is it numeric 0 or the zero vector? (MATLAB distinguishes.)

Quick Reference Formula Sheet

• $\mathbf a\cdot\mathbf b = \sum<em>{i=1}^{n} a</em>i b_i.$

• $\cos\theta = \dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|\,|\mathbf b|}.$

• $\operatorname{comp}_{\mathbf a}\mathbf b = \dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|}.$

• $\operatorname{proj}_{\mathbf a}\mathbf b = \left(\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|^{2}}\right)\mathbf a.$

• $\mathbf a\times\mathbf b = (a<em>2 b</em>3-a<em>3 b</em>2,\;a<em>3 b</em>1-a<em>1 b</em>3,\;a<em>1 b</em>2-a<em>2 b</em>1).$

• $|\mathbf a\times\mathbf b| = |\mathbf a|\,|\mathbf b|\,\sin\theta.$

• $\text{Area}_{\text{parallelogram}} = |\mathbf a\times\mathbf b|.$

Suggested Practice

1. Compute angles between random 3-D vectors using dot-product formula.

2. Given a triangle in $\mathbb R^3$, find normal vector via cross product.

3. Verify Cauchy–Schwarz numerically on several vector pairs.

4. Decompose a force vector into components parallel and perpendicular to a surface normal.

Happy studying – master these tools; they recur in multivariable calculus, physics, computer graphics, and beyond!