Vectors & Scalars – Comprehensive Study Notes

Scalars vs Vectors

• Scalar ("Scaler")
  • Described completely by magnitude only.
  • Do not contain directional information.
  • Examples: Work, speed, distance, mass, temperature, density, volume.
• Vector ("Dector")
  • Possess both magnitude and direction.
  • Denoted graphically by an arrow: tail → head.
  • Written with a bar/arrow overhead, boldface, or component form, e.g. $\vec A$, A, $A<em>x \hat\imath + A</em>y \hat\jmath$.
  • Examples: Force, velocity, displacement, weight, momentum, electric field.
  • Sample description: “Vector = 4 units South”, “$\vec F = 100\,\text{N}$ toward +x”.

Physical vs Non-Physical Quantities

• Physical quantity – measurable; can be expressed numerically with a unit.
  • Examples: Mass, velocity, temperature.
• Non-physical quantity – cannot be measured directly with instruments.
  • Examples: Love, hate, goodness, inertia (as a qualitative idea).

Representation & Terminology

• Magnitude (length) of $\vec A$ is $|\vec A|=A$.
• Tail: starting point; Head: arrow point.
• Position vector $\vec r$: directed from chosen origin to position of object.
• Displacement vector $\Delta \vec r$: change in position in a given interval; “how much & which way.”

Unit Vectors (Direction Cosines)

• Have unit magnitude (1) & no physical unit; specify direction only.
• Standard orthogonal (Cartesian) basis
  • $\hat\imath$: +x direction  $\hat\jmath$: +y  $\hat k$: +z
• Any vector expressed as
  $\boxed{\vec A = A<em>x \hat\imath + A</em>y \hat\jmath + A_z \hat k}$
• Components obtained by
  $A<em>x = A\cos\alpha,\; A</em>y = A\cos\beta,\; A_z = A\cos\gamma$
  where $\alpha,\beta,\gamma$ are angles with x-, y-, z-axes.

Classes of Vectors

• Equal Vectors: same magnitude & same direction.
• Parallel Vectors: direction identical or opposite; magnitudes may differ.
• Negative Vector: same magnitude as reference but opposite direction ($\vec A$ and $-\vec A$).
• Anti-parallel: simply opposite directions; magnitudes may differ.
• Collinear: lie on same straight line.
• Coplanar: lie in the same plane.
• Orthogonal: angle $90^\circ$ between them.
• Concurrent: lines of action intersect at one point.
• Axial (pseudo) vector: direction along axis of rotation (e.g.
  $\vec L = \vec r \times \vec p$).
• Zero/Null vector: magnitude $0$; no defined direction.
• Unit vector: magnitude $1$, serves only for direction.

Head-to-Tail, Triangle & Parallelogram Laws

• Triangle Law: If $\vec A$ & $\vec B$ are placed head-to-tail in order, resultant $\vec R = \vec A + \vec B$ is the third side taken from tail of first to head of last.
• Parallelogram Law: Place $\vec A$ & $\vec B$ tail-to-tail; $\vec R$ is diagonal of parallelogram.
• Magnitude of resultant (general):
  $\boxed{R = \sqrt{A^2 + B^2 + 2AB\cos\theta}}$ where $\theta$ is the smaller angle between $\vec A$ & $\vec B$.
• Direction (angle $\phi$ that $\vec R$ makes with $\vec A$):
  $\boxed{\tan\phi = \dfrac{B\sin\theta}{A + B\cos\theta}}$
• Commutative: $\vec A + \vec B = \vec B + \vec A$.

Subtraction of Vectors

• $\vec A - \vec B = \vec A + (-\vec B)$; reverse $\vec B$ then add.
• If angle between $\vec A$ and $\vec B$ is $\theta$, angle between $\vec A$ and $-\vec B$ is $(180^\circ-\theta)$.
• Magnitude of difference:
  $R = \sqrt{A^2 + B^2 - 2AB\cos\theta}$.

Special Kinematics Example – Turning Vehicle/Aircraft

• Object speed $v$ turns through angle $\theta$ but maintains speed; change in velocity magnitude:
  $\boxed{\Delta v = 2v\sin\dfrac{\theta}{2}}$
  (derived from vector subtraction of equal-length vectors forming isosceles triangle).
• Example: $v=20\,\text{m/s},\ \theta = 60^\circ\;\Rightarrow\; \Delta v = 20\,\text{m/s}.$

Resolution of a Vector

• In 2-D: $\vec F = F<em>x \hat\imath + F</em>y \hat\jmath$ with
  $F<em>x = F\cos\theta,\; F</em>y = F\sin\theta$ ((\theta) measured from +x).
• Magnitude check: $F = \sqrt{F<em>x^2 + F</em>y^2}$.
• 3-D magnitude: $|\vec A| = \sqrt{A<em>x^2 + A</em>y^2 + A_z^2}$.
• Example: $|\vec A| =10$ at $30^\circ$ above +x axis → $A<em>x = 10\cos30^\circ = 5\sqrt3,\; A</em>y = 10\sin30^\circ = 5$.

Unit Vector Along a Given Vector

• $\displaystyle \hat u_{A}=\frac{\vec A}{|\vec A|}.$
• E.g.
  $\vec A = 3\hat\imath + 4\hat\jmath \;\Rightarrow\; |\vec A| = 5,\; \hat u_{A}=\dfrac{3\hat\imath + 4\hat\jmath}{5}.$

Dot (Scalar) Product

• Definition: $\boxed{\vec A \cdot \vec B = |\vec A||\vec B|\cos\theta = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z}$.
• Properties
  • Scalar result.
  • Commutative: $\vec A\cdot\vec B = \vec B\cdot\vec A$.
  • $\vec A\cdot\vec A = |\vec A|^2$.
  • Orthogonality: $\vec A\cdot\vec B = 0 \iff \theta = 90^\circ$.
• Physical example: Work done $W = \vec F \cdot \vec s$.
  • If $F = 10\,\text N,\ s=5\,\text m,\ \theta = 60^\circ$ → $W = 10\times 5\cos60^\circ = 25\,\text J$.

Cross (Vector) Product

• Definition: $\boxed{\vec A \times \vec B = |\vec A||\vec B|\sin\theta\; \hat n}$
  where $\hat n$ is a unit vector perpendicular to the plane of $\vec A,\vec B$ following right-hand rule.
• Magnitude zero when vectors are parallel ($\theta=0,\,180^\circ$).
• Direction relations for basis vectors:
  $\hat\imath\times \hat\jmath = \hat k,\; \hat\jmath\times \hat k = \hat\imath,\; \hat k \times \hat\imath = \hat\jmath$. Reverse order changes sign.
• Determinant form (components):
  \vec A\times\vec B =
  \begin{vmatrix}
  \hat\imath & \hat\jmath & \hat k\
  Ax & Ay & Az\ Bx & By & Bz
  \end{vmatrix}.
• Example: $\vec A = 2\hat\imath + 3\hat\jmath + 4\hat k,\; \vec B = 3\hat\imath + 2\hat\jmath + 3\hat k$
  • $\vec A \times \vec B = (3\cdot3 - 4\cdot2)\hat\imath + (4\cdot3 - 2\cdot3)\hat\jmath + (2\cdot2 - 3\cdot3)\hat k = (1)\hat\imath + (6)\hat\jmath + (-5)\hat k.$

Angle Between Two Vectors

• From dot product: $\displaystyle \cos\theta = \frac{\vec A\cdot\vec B}{|\vec A||\vec B|}.$
• From cross product: $\displaystyle \sin\theta = \frac{|\vec A\times\vec B|}{|\vec A||\vec B|}.$
• Example: If difference of two unit vectors is also a unit vector → $1 = 1 + 1 - 2\cos\theta \Rightarrow \cos\theta = \tfrac12 \Rightarrow \theta = 60^\circ.$
• If sum and difference have equal magnitude: implies $\theta = 90^\circ.$

Concurrent & Resultant – Worked Examples

• Two forces $6\,\text N$ each at $60^\circ$:
  $R = \sqrt{6^2 + 6^2 + 2(6)(6)\cos60^\circ} = 6\sqrt3\,\text N.$
• Component form: $\vec F<em>1=6\hat\imath,\; \vec F</em>2=6\cos60^\circ\hat\imath + 6\sin60^\circ\hat\jmath,$ then add.
• Turning car: initial $\vec v<em>1 = 5\hat\imath,$ final $\vec v</em>2 = 5\hat\jmath$ → $\Delta v = 5\sqrt2$ toward NW.

Electrostatics Example (Central Force)

• For electron in orbit radius $r$ around proton:
  $\vec F = k \frac{q<em>1 q</em>2}{r^2}\,\hat r$ (directed radially toward centre).

Laws from Elementary Geometry

• Cosine law (triangle ABC, side a opposite $\alpha$):
  $\boxed{a^2 = b^2 + c^2 - 2bc\cos\alpha}$.
  Directly yields magnitude formula for $\vec R = \vec B + \vec C$.
• Sine law: $\displaystyle \frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}.$
• Useful for determining unknown sides/angles in vector polygons.

Quick Reference – Key Formulae

• Resultant magnitude (addition): $R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$.
• Resultant magnitude (subtraction): $R = \sqrt{A^2 + B^2 - 2AB\cos\theta}$.
• Direction of resultant: $\tan\phi = \dfrac{B\sin\theta}{A + B\cos\theta}$.
• Change in velocity during turn: $\Delta v = 2v\sin\dfrac{\theta}{2}$.
• Scalar product: $\vec A\cdot\vec B = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$.
• Vector product magnitude: $|\vec A\times\vec B| = |\vec A||\vec B| \sin\theta$.
• Cartesian to magnitude: $|\vec A| = \sqrt{A<em>x^2 + A</em>y^2 + A_z^2}$.
• Unit vector: $\hat u = \dfrac{\vec A}{|\vec A|}$.

These bullet-point notes encapsulate every major and minor detail from the transcript: definitions, classifications, graphical conventions, head–tail operations, derived formulas, geometric proofs, example calculations, orthogonal unit-vector algebra, dot & cross products, change-in-velocity applications, and the sine/cosine laws that underpin resultant derivations.