Summary

Vectors — the basics.

A scalar has magnitude only; a vector has magnitude and direction. Two vectors are equal

only if both magnitude and direction match. Multiplying a vector by a scalar $\lambda$

scales its magnitude by $|\lambda|$ and reverses its direction if $\lambda < 0$; the unit

vector $\hat A = \vec A/A$ has magnitude 1 and gives pure direction.

Addition, subtraction, resolution.

Vectors add by the triangle/parallelogram law:

$R = \sqrt{A^2+B^2+2AB\cos\theta}$, $\tan\alpha = B\sin\theta/(A+B\cos\theta)$. Subtraction is

addition of the negative vector. A vector resolves into rectangular components

$A_x = A\cos\theta$, $A_y = A\sin\theta$, with $A = \sqrt{A_x^2+A_y^2}$. In component form,

$\vec A = A_x\hat i + A_y\hat j$, and vectors add/subtract by adding/subtracting components.

Scalar and vector products.

$\vec A \cdot \vec B = AB\cos\theta = A_xB_x + A_yB_y$ (a scalar; zero when perpendicular;

used for work, $W = \vec F \cdot \vec d$). $\vec A \times \vec B = AB\sin\theta\,\hat n$

(right-hand rule; a vector, magnitude = area of the parallelogram spanned by $\vec A$,

$\vec B$; zero when parallel; used for torque, $\vec\tau = \vec r \times \vec F$).

Relative velocity. $\vec v_{AB} = \vec v_A - \vec v_B$, found by vector subtraction, not

by subtracting speeds — used in river-crossing and rain-and-walker problems.

Motion in a plane. $\vec v = \vec v_0 + \vec a t$, $\vec r = \vec r_0 + \vec v_0 t + \tfrac12\vec a t^2$; each resolves into two independent 1-D equations along x and y

(principle of independence of motion). …