Vectors

A vector has magnitude and direction: $\vec{a} = (a_1, a_2, a_3)$, $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$. Physics runs on two products of vectors — one gives a scalar, one gives a vector.

Dot product

$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\vec{a}|\,|\vec{b}|\cos\theta$

• Scalar result.
• $\vec{a} \cdot \vec{b} = 0 \iff$ perpendicular.
• Projection of $\vec{a}$ onto $\vec{b}$: $\dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.
• Physics: work $W = \vec{F} \cdot \vec{s}$.

Worked example. Angle between $\vec{a} = (1, 2, 2)$ and $\vec{b} = (2, -2, 1)$:

$\vec{a} \cdot \vec{b} = 2 - 4 + 2 = 0 \Rightarrow \theta = 90°$

No cosine computation needed — a zero dot product ends the discussion.

Exercise 3.1

Given $\vec{u} = (3, -1, 2)$ and $\vec{v} = (1, 4, -2)$:

a) Compute $\vec{u} \cdot \vec{v}$ and $|\vec{u}|$, $|\vec{v}|$.

b) Compute $\cos\theta$ and state whether $\theta$ is acute or obtuse.

c) Find the projection of $\vec{u}$ onto $\vec{v}$ (scalar projection).

d) Find a nonzero vector perpendicular to $\vec{u}$ (any valid one).

Cross product

$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{e}_1 & \vec{e}_2 & \vec{e}_3 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2 b_3 - a_3 b_2,\; a_3 b_1 - a_1 b_3,\; a_1 b_2 - a_2 b_1)$

• Vector result, perpendicular to both factors; direction by right-hand rule.
• $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ = area of the parallelogram they span.
• Anticommutative: $\vec{b} \times \vec{a} = -\,\vec{a} \times \vec{b}$.
• Physics: torque $\vec{\tau} = \vec{r} \times \vec{F}$, Lorentz force $\vec{F} = q\vec{v} \times \vec{B}$.

Worked example. $\vec{a} = (1, 0, 2)$, $\vec{b} = (0, 3, -1)$:

$\vec{a} \times \vec{b} = (0 \cdot (-1) - 2 \cdot 3,\; 2 \cdot 0 - 1 \cdot (-1),\; 1 \cdot 3 - 0 \cdot 0) = (-6, 1, 3)$

Check: $(-6,1,3) \cdot (1,0,2) = -6 + 0 + 6 = 0$. ✓

Exercise 3.2

Given $\vec{u} = (2, 1, 0)$ and $\vec{v} = (1, -1, 3)$:

a) Compute $\vec{u} \times \vec{v}$.

b) Verify your result is perpendicular to $\vec{u}$ (dot product).

c) Area of the parallelogram spanned by $\vec{u}$ and $\vec{v}$.

d) Compute $\vec{v} \times \vec{u}$ without redoing the determinant.

Geometric interpretation

Everything above in one picture: the dot product measures alignment (projection), the cross product measures spanned area and gives the normal.

import numpy as np
import matplotlib.pyplot as plt

u = np.array([3.0, 1.0])   # edit me
v = np.array([1.0, 2.5])   # edit me

proj = (u @ v) / (v @ v) * v            # vector projection of u onto v
area = abs(u[0]*v[1] - u[1]*v[0])       # |cross| in 2D

ax = plt.gca()
for vec, c, name in [(u,'C0','u'), (v,'C1','v'), (proj,'C2','proj_v u')]:
    ax.quiver(0, 0, vec[0], vec[1], angles='xy', scale_units='xy', scale=1, color=c, label=name)
ax.plot([u[0], proj[0]], [u[1], proj[1]], 'k:')
ax.set_xlim(-1, 4); ax.set_ylim(-1, 4); ax.set_aspect('equal')
ax.legend(); ax.grid(True)
ax.set_title(f'u·v = {u@v:.2f}   parallelogram area = {area:.2f}')
plt.show()

Exercise 3.3

Work these geometrically — sketch by hand where it helps:

a) The points $A = (1,0,0)$, $B = (0,2,0)$, $C = (0,0,3)$ form a triangle. Compute its area via a cross product.

b) For which value of $t$ are $(t, 2, 1)$ and $(4, t, -2)$ perpendicular?

c) A force $\vec{F} = (5, 0, 0)$ N acts on a lever arm $\vec{r} = (0.2, 0.3, 0)$ m. Compute the torque $\vec{\tau} = \vec{r} \times \vec{F}$ and interpret its direction physically.