Unit 1: Units, Dimensions & Vectors

This is the foundational unit of JEE Physics. Although seemingly elementary, it appears every year in JEE Main (1–2 questions, ~8 marks) and frequently in JEE Advanced as part of a multi-concept question or a Match-the-Column. Mastery of dimensions, errors and vectors is a prerequisite for every later unit — kinematics, electrodynamics and modern physics all rely on vector calculus and the ability to detect dimensionally inconsistent options under exam pressure.

Typical question types:

• JEE Main: "Find the dimensional formula of …", "If $a=bc/d$ and the percentage errors in $b, c, d$ are …, find the maximum % error in $a$", "Resultant of two vectors", "Angle between $\vec A$ and $\vec B$ given $\vec A\cdot\vec B$ and $\vert \vec A\times\vec B\vert$".
• JEE Advanced: Dimensional formulas of unfamiliar constants (e.g. coefficient in a given empirical equation), trap problems combining dimensions with error analysis, vector geometry in 3D, scalar triple product interpretation.

Concept Map

• Units & Measurement
  • SI base units (7), supplementary units (radian, steradian), derived units
  • Practical units (light-year, parsec, AU, atomic mass unit, electron-volt, bar, torr)
  • Conversion of units across systems
• Dimensional Analysis
  • Dimensional formula of mechanical, thermal, electromagnetic quantities
  • Principle of homogeneity
  • Deducing formulae (pendulum, viscous force, drag, Bohr radius)
  • Limitations of dimensional analysis
• Significant Figures & Error Analysis
  • Rules for significant figures and rounding
  • Absolute, relative and percentage error
  • Propagation in $+, -, \times, \div, x^n$
  • Combination of independent errors (max & probable)
• Vectors
  • Scalars vs vectors; equal, negative, null, unit, position, displacement
  • Triangle, parallelogram, polygon laws; addition, subtraction
  • Resolution into rectangular components in 2D and 3D
  • Direction cosines, direction ratios
• Vector Products
  • Dot (scalar) product: definition, properties, geometric meaning
  • Cross (vector) product: definition, right-hand rule, area interpretation
  • Scalar triple product (volume), vector triple product (mention)

Topic 1: Units and Measurement

Sub-topic A: SI System

The SI system has 7 base units:

Two supplementary units (now subsumed under derived in modern SI):

• Plane angle: radian (rad)
• Solid angle: steradian (sr)

A solid angle subtended at the centre of a sphere of radius $r$ by area $A$ on its surface is $\Omega = A/r^2$. The full sphere subtends $4\pi$ sr.

Sub-topic B: Practical units

• $1\,\text{AU} = 1.496\times10^{11}$ m (mean Earth–Sun distance)
• $1\,\text{light-year} = 9.46\times10^{15}$ m
• $1\,\text{parsec} = 3.08\times10^{16}$ m $= 3.26$ ly
• $1\,\text{u} = 1.66\times10^{-27}$ kg (atomic mass unit)
• $1\,\text{eV} = 1.602\times10^{-19}$ J
• $1\,\text{bar} = 10^5$ Pa, $1\,\text{atm} = 1.013\times10^5$ Pa, $1\,\text{torr} = 133.3$ Pa

Sub-topic C: Conversion of units

A physical quantity has the same magnitude in all unit systems when expressed dimensionally; i.e.

$n_1 [M_1^a L_1^b T_1^c] = n_2 [M_2^a L_2^b T_2^c]$

so

$n_2 = n_1 \left(\frac{M_1}{M_2}\right)^a \left(\frac{L_1}{L_2}\right)^b \left(\frac{T_1}{T_2}\right)^c.$

Worked Examples (JEE Main level)

Example 1.1. Convert $1$ joule into ergs using dimensional analysis.

$[J] = M L^2 T^{-2}$. In SI, $M_1=1$ kg, $L_1=1$ m, $T_1=1$ s. In CGS, $M_2=1$ g, $L_2=1$ cm, $T_2=1$ s.

$n_2 = 1 \cdot \left(\frac{1000\,\text{g}}{1\,\text{g}}\right)^1 \left(\frac{100\,\text{cm}}{1\,\text{cm}}\right)^2 \left(\frac{1}{1}\right)^{-2} = 10^3 \cdot 10^4 = 10^7.$

$1\,\text{J} = 10^7\,\text{erg}$.

Example 1.2. The density of mercury is $13.6\,\text{g/cm}^3$. Convert it into kg/m³.

$[\rho] = ML^{-3}$. With $M_1$=g, $L_1$=cm and target $M_2$=kg, $L_2$=m:

$n_2 = 13.6 \cdot \frac{10^{-3}\,\text{kg}}{1\,\text{kg}} \cdot \left(\frac{10^{-2}\,\text{m}}{1\,\text{m}}\right)^{-3} = 13.6\cdot 10^{-3}\cdot 10^{6} = 1.36\times10^{4}\,\text{kg/m}^3.$

Topic 2: Dimensional Analysis

Sub-topic A: Dimensional formulae of common quantities

Sub-topic B: Principle of homogeneity

Every additive term and every side of a physical equation must have the same dimensions. This is used to:

1. Check correctness of an equation.
2. Convert units.
3. Deduce relations among physical quantities (up to a dimensionless constant).

Sub-topic C: Deduction of formulae

Example: Period of a simple pendulum. Suppose $T$ depends on length $\ell$, mass $m$, gravity $g$:

$T = k\,\ell^a m^b g^c.$

Dimensionally: $T^1 = L^a M^b (LT^{-2})^c = M^b L^{a+c} T^{-2c}$.

Equating exponents: $b=0$, $a+c=0$, $-2c=1 \Rightarrow c=-1/2$, $a=1/2$. So $T=k\sqrt{\ell/g}$. (Experiment gives $k=2\pi$.)

Example: Stokes' viscous force. Assume $F$ on a sphere of radius $r$ moving with velocity $v$ in a fluid of viscosity $\eta$ is

$F=k\,\eta^a r^b v^c.$

$[F]=MLT^{-2}$, $[\eta]=ML^{-1}T^{-1}$, $[r]=L$, $[v]=LT^{-1}$.

$MLT^{-2}=M^a L^{-a+b+c}T^{-a-c}\Rightarrow a=1,\;-a-c=-2\Rightarrow c=1,\;-a+b+c=1\Rightarrow b=1.$ So $F=k\eta r v$, with $k=6\pi$ from experiment.

Example: Bohr radius. Estimate the size of a hydrogen atom from $h$, $m_e$, $e$, $\varepsilon_0$:

$a_0 = k\,\frac{\varepsilon_0^a h^b}{m_e^c e^d}.$

A dimensional check yields $a_0 = \frac{\varepsilon_0 h^2}{\pi m_e e^2}\approx 0.529$ Å.

Sub-topic D: Limitations

Dimensional analysis cannot:

• determine dimensionless constants ($\pi, e$, numbers like 2, 1/2),
• distinguish quantities with the same dimensions (work vs torque, angular momentum vs Planck's constant),
• handle equations containing functions like $\sin x$, $e^x$, $\log x$ (their arguments must be dimensionless),
• handle quantities depending on more than three mechanical quantities (M, L, T) — over-determined.

Worked Examples (JEE Main level)

Example 2.1. Find $[a]$ and $[b]$ in $\left(P+\dfrac{a}{V^2}\right)(V-b)=RT$ (van der Waals).

Since $a/V^2$ must have the dimensions of pressure: $[a]=[P][V^2]= ML^{-1}T^{-2}\cdot L^6 = ML^{5}T^{-2}$. And $[b]=[V]=L^3$.

Example 2.2. The dimensions of $\dfrac{1}{\mu_0\varepsilon_0}$.

$[\mu_0\varepsilon_0]=[1/c^2]=T^2L^{-2}\Rightarrow[1/\mu_0\varepsilon_0]=L^2T^{-2}$ — square of speed. Confirms $c=1/\sqrt{\mu_0\varepsilon_0}$.

Example 2.3. The position of a particle is $x=at+bt^2-ct^3$. Find dimensions of $a, b, c$.

Each term has units of length: $[a]=LT^{-1}$, $[b]=LT^{-2}$, $[c]=LT^{-3}$.

Worked Examples (JEE Advanced level)

Example 2.A1. In a given system, mass is measured in units $\alpha$ kg, length in $\beta$ m and time in $\gamma$ s. The unit of force in this system, expressed in newtons, is:

A force of $1$ N has dimensions $MLT^{-2}$. Number of new units of force in $1$ N is $n_2 = 1\cdot(1/\alpha)\cdot(1/\beta)\cdot(1/\gamma)^{-2} = \gamma^2/(\alpha\beta)$. Hence $1$ unit of the new force $= \alpha\beta/\gamma^2$ N.

Example 2.A2. The frequency of vibration of a stretched string depends on its length $L$, tension $T$ and mass per unit length $\mu$. Derive the formula.

$\nu = k L^a T^b \mu^c$. $[\nu]=T^{-1}$, $[L]=L$, $[T]=MLT^{-2}$, $[\mu]=ML^{-1}$.

$T^{-1}= L^a (MLT^{-2})^b (ML^{-1})^c = M^{b+c}L^{a+b-c}T^{-2b}.$

So $b+c=0$, $a+b-c=0$, $-2b=-1\Rightarrow b=1/2$, $c=-1/2$, $a=-1$.

$\boxed{\nu = \frac{k}{L}\sqrt{\frac{T}{\mu}}}.$

Experimentally $k=1/2$ for the fundamental.

Topic 3: Significant Figures, Rounding and Error Analysis

Sub-topic A: Rules for significant figures

1. All non-zero digits are significant.
2. All zeros between non-zero digits are significant ($1002 \to 4$ s.f.).
3. Leading zeros are NOT significant ($0.0023\to 2$ s.f.).
4. Trailing zeros after a decimal point are significant ($2.300\to 4$ s.f.).
5. Trailing zeros in an integer without a decimal point are ambiguous. Use scientific notation: $4500$ could be $2, 3$ or $4$ s.f.; write $4.500\times10^3$ for $4$ s.f.
6. In addition/subtraction, the result has the same number of decimal places as the operand with the fewest.
7. In multiplication/division, the result has the same number of significant figures as the operand with the fewest.

Sub-topic B: Rounding rules

Round half to even (banker's rounding) is recommended; in NCERT-style: round-half-up is acceptable. $2.345 \to 2.34$ or $2.35$ depending on convention — JEE typically accepts either.

Sub-topic C: Errors — definitions

• Absolute error $\Delta x = \vert x_{\text{measured}} - x_{\text{true}}\vert$. When the true value is unknown, use $\bar x = \tfrac1n\sum x_i$ and $\Delta x_i = \vert x_i - \bar x\vert$.
• Mean absolute error $\overline{\Delta x} = \tfrac1n\sum \vert \Delta x_i\vert$.
• Relative error $= \overline{\Delta x}/\bar x$.
• Percentage error $= (\overline{\Delta x}/\bar x)\times 100\%$.

Sub-topic D: Propagation of errors

Suppose $A, B$ have absolute errors $\Delta A, \Delta B$.

Sum / Difference. $Z = A\pm B \Rightarrow \Delta Z = \Delta A + \Delta B$ (errors add). Relative error in a difference can blow up if $A\approx B$ — exam favourite trap.

Product / Quotient. $Z = A B$ or $A/B$:

$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}.$

Power. $Z = A^p B^q / C^r$:

$\frac{\Delta Z}{Z} = |p|\frac{\Delta A}{A} + |q|\frac{\Delta B}{B} + |r|\frac{\Delta C}{C}.$

Derivation of the power rule. Take logs: $\ln Z = p\ln A + q\ln B - r\ln C$. Differentiate: $dZ/Z = p\,dA/A + q\,dB/B - r\,dC/C$. The maximum (worst-case) error replaces $d\to \Delta$ and signs by $\vert \cdot\vert$, giving the formula above.

Worked Examples (JEE Main level)

Example 3.1. A physical quantity $X = \dfrac{a^2 b^3}{c\sqrt{d}}$. Given % errors $\Delta a/a = 1\%$, $\Delta b/b = 2\%$, $\Delta c/c = 3\%$, $\Delta d/d = 4\%$. Find the maximum % error in $X$.

$\frac{\Delta X}{X} = 2(1\%) + 3(2\%) + 1(3\%) + \tfrac12(4\%) = 2+6+3+2 = 13\%.$

Example 3.2. Two resistors of $100\pm 3\,\Omega$ and $200\pm 4\,\Omega$ are connected in series. Find equivalent resistance with absolute uncertainty.

$R = 300\,\Omega$, $\Delta R = 3+4 = 7\,\Omega$.

Example 3.3. Same resistors in parallel.

$R_p = (100)(200)/300 = 66.\overline 6\,\Omega$. Using $1/R_p = 1/R_1+1/R_2$, $\Delta R_p/R_p^2 = \Delta R_1/R_1^2 + \Delta R_2/R_2^2$ so

$\Delta R_p = R_p^2 \left(\frac{3}{100^2}+\frac{4}{200^2}\right) = (66.7)^2 (3\cdot10^{-4} + 10^{-4}) \approx 1.78\,\Omega.$

Worked Examples (JEE Advanced level)

Example 3.A1. A screw gauge has pitch $0.5$ mm and $50$ divisions on the circular scale. A wire reading is main scale $3$ mm, circular scale $35$. Find the diameter and its uncertainty assuming a half-division least count.

Least count $= 0.5/50 = 0.01$ mm. Diameter $= 3 + 35\times0.01 = 3.35$ mm. Uncertainty $\pm 0.005$ mm.

Example 3.A2. Young's modulus is determined from $Y = \dfrac{FL}{A\Delta L} = \dfrac{(Mg) L}{(\pi d^2/4)\Delta L}$. Given $M=2.00\pm0.01$ kg, $L=1.000\pm0.001$ m, $d=0.500\pm0.005$ mm, $\Delta L = 0.250\pm0.005$ mm. Maximum % error in $Y$?

$\frac{\Delta Y}{Y} = \frac{\Delta M}{M} + \frac{\Delta L}{L} + 2\frac{\Delta d}{d} + \frac{\Delta(\Delta L)}{\Delta L}$

$= 0.5\% + 0.1\% + 2(1\%) + 2\% = 4.6\%$.

Topic 4: Vectors — Algebra

Sub-topic A: Definitions

• Scalar: magnitude only (mass, time, temperature, work, energy, power, pressure, charge, frequency, current).
• Vector: magnitude + direction + obeys parallelogram law (displacement, velocity, acceleration, force, momentum, torque, angular velocity, electric field, magnetic field).
• Trap: current is a scalar in spite of having a direction in a wire (it does not satisfy the parallelogram law at a junction).

A vector $\vec A$ has magnitude $\vert \vec A\vert =A$. Unit vector: $\hat A = \vec A / A$.

Sub-topic B: Triangle and parallelogram laws

If $\vec R = \vec A + \vec B$ with angle $\theta$ between $\vec A$ and $\vec B$:

$R = \sqrt{A^2+B^2+2AB\cos\theta}.$

Direction (angle $\alpha$ of $\vec R$ from $\vec A$):

$\tan\alpha = \frac{B\sin\theta}{A+B\cos\theta}.$

Polygon law generalizes for any number of coplanar vectors taken head to tail; the closing side gives the resultant (with direction reversed).

Special cases:

• $\theta=0\Rightarrow R=A+B$ (max).
• $\theta=180^\circ\Rightarrow R=\vert A-B\vert$ (min).
• $\theta=90^\circ\Rightarrow R=\sqrt{A^2+B^2}$.
• If $A=B$, $R=2A\cos(\theta/2)$, $\alpha=\theta/2$.

Sub-topic C: Resolution into components

In 2D: $\vec A = A_x\hat\imath + A_y\hat\jmath$, where $A_x=A\cos\theta$, $A_y=A\sin\theta$.

In 3D: $\vec A = A_x\hat\imath + A_y\hat\jmath + A_z\hat k$. Magnitude $A=\sqrt{A_x^2+A_y^2+A_z^2}$. Direction cosines:

$\cos\alpha=\frac{A_x}{A},\quad\cos\beta=\frac{A_y}{A},\quad\cos\gamma=\frac{A_z}{A},\quad \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1.$

Sub-topic D: Position and displacement vectors

For a particle at $\vec r(t)=x\hat\imath+y\hat\jmath+z\hat k$ with respect to origin $O$, displacement between $t_1$ and $t_2$ is $\Delta\vec r=\vec r(t_2)-\vec r(t_1)$.

Worked Examples (JEE Main level)

Example 4.1. Two forces of $5$ N and $12$ N act at right angles. Find resultant and direction.

$R=\sqrt{25+144}=13$ N. $\tan\alpha = 12/5\Rightarrow \alpha=67.4^\circ$ from the 5 N force.

Example 4.2. If $\vec A+\vec B = \vec C$ and $A=3, B=4, C=5$, the angle between $\vec A$ and $\vec B$ is?

$C^2=A^2+B^2+2AB\cos\theta\Rightarrow 25=9+16+24\cos\theta\Rightarrow\cos\theta=0\Rightarrow\theta=90^\circ$.

Example 4.3. A vector $\vec A=2\hat\imath+3\hat\jmath-6\hat k$. Find a unit vector parallel to $\vec A$.

$\vert \vec A\vert =\sqrt{4+9+36}=7$. $\hat A=(2\hat\imath+3\hat\jmath-6\hat k)/7$.

Worked Examples (JEE Advanced level)

Example 4.A1. Two vectors of equal magnitude $A$ have a resultant of magnitude $A\sqrt 3$. Find the angle between them.

$3A^2 = 2A^2 + 2A^2\cos\theta \Rightarrow \cos\theta = 1/2\Rightarrow \theta=60^\circ$.

Example 4.A2. A particle is displaced from $(2,1,3)$ m to $(-1,4,5)$ m. Find magnitude of displacement and its angle with the $x$-axis.

$\Delta\vec r=(-3,3,2)$ m. $\vert \Delta\vec r\vert =\sqrt{9+9+4}=\sqrt{22}$ m. $\cos\alpha=-3/\sqrt{22}\Rightarrow \alpha\approx129.8^\circ$.

Topic 5: Dot Product and Cross Product

Sub-topic A: Dot product

$\vec A\cdot\vec B = AB\cos\theta = A_xB_x+A_yB_y+A_zB_z.$

Properties:

• Commutative: $\vec A\cdot\vec B=\vec B\cdot\vec A$.
• Distributive: $\vec A\cdot(\vec B+\vec C)=\vec A\cdot\vec B+\vec A\cdot\vec C$.
• $\hat\imath\cdot\hat\imath=1$, $\hat\imath\cdot\hat\jmath=0$.
• $\vec A\cdot\vec A=A^2$.
• Component of $\vec A$ along $\vec B$: $\vec A\cdot\hat B = A\cos\theta$.
• Geometric: projection of one vector on another.

Applications: work $W=\vec F\cdot\vec d$; power $P=\vec F\cdot\vec v$; magnetic flux $\Phi=\vec B\cdot\vec A$.

Sub-topic B: Cross product

$\vec A\times\vec B = AB\sin\theta\,\hat n,$

where $\hat n$ is perpendicular to the plane of $\vec A, \vec B$ by the right-hand rule.

In Cartesian form,

$\vec A\times\vec B=\begin{vmatrix}\hat\imath&\hat\jmath&\hat k\\A_x&A_y&A_z\\B_x&B_y&B_z\end{vmatrix}=(A_yB_z-A_zB_y)\hat\imath-(A_xB_z-A_zB_x)\hat\jmath+(A_xB_y-A_yB_x)\hat k.$

Properties:

• Anti-commutative: $\vec A\times\vec B=-\vec B\times\vec A$.
• Distributive.
• $\hat\imath\times\hat\jmath=\hat k$, $\hat\jmath\times\hat k=\hat\imath$, $\hat k\times\hat\imath=\hat\jmath$ (cyclic).
• $\vec A\times\vec A=\vec 0$.
• $\vert \vec A\times\vec B\vert$ = area of parallelogram with $\vec A, \vec B$ as adjacent sides; half of it = area of triangle.

Applications: torque $\vec\tau=\vec r\times\vec F$; angular momentum $\vec L=\vec r\times\vec p$; magnetic force $\vec F=q\vec v\times\vec B$.

Sub-topic C: Scalar and vector triple products

• Scalar triple product: $[\vec A\,\vec B\,\vec C] = \vec A\cdot(\vec B\times\vec C)$ = volume of parallelepiped.
• Cyclic invariance: $[\vec A\,\vec B\,\vec C]=[\vec B\,\vec C\,\vec A]=[\vec C\,\vec A\,\vec B]$.
• $[\vec A\,\vec B\,\vec C]=0\Leftrightarrow$ coplanar.
• Vector triple product: $\vec A\times(\vec B\times\vec C)=(\vec A\cdot\vec C)\vec B-(\vec A\cdot\vec B)\vec C$ — the BAC-CAB rule.

Worked Examples (JEE Main level)

Example 5.1. $\vec A=2\hat\imath+\hat\jmath-2\hat k$, $\vec B=\hat\imath-\hat\jmath+2\hat k$. Find $\vec A\cdot\vec B$ and the angle.

$\vec A\cdot\vec B=2-1-4=-3$. $A=3,\,B=\sqrt{1+1+4}=\sqrt 6$. $\cos\theta=-3/(3\sqrt 6)=-1/\sqrt 6\Rightarrow\theta\approx114^\circ$.

Example 5.2. Find a unit vector perpendicular to both $\vec A=\hat\imath-\hat\jmath$ and $\vec B=\hat\jmath+\hat k$.

$\vec A\times\vec B=\begin{vmatrix}\hat\imath&\hat\jmath&\hat k\\1&-1&0\\0&1&1\end{vmatrix}=\hat\imath(-1-0)-\hat\jmath(1-0)+\hat k(1-0)=-\hat\imath-\hat\jmath+\hat k$. Magnitude $\sqrt 3$, so $\hat n=(-\hat\imath-\hat\jmath+\hat k)/\sqrt 3$.

Example 5.3. A constant force $\vec F=2\hat\imath+3\hat\jmath+4\hat k$ N acts on a particle, displacing it from $(1,1,1)$ to $(2,3,4)$ m. Work done?

$\vec d=(1,2,3)$ m. $W=\vec F\cdot\vec d=2+6+12=20$ J.

Worked Examples (JEE Advanced level)

Example 5.A1. If $\vert \vec A+\vec B\vert =\vert \vec A-\vec B\vert$, show $\vec A\perp\vec B$.

$\vert \vec A+\vec B\vert ^2-\vert \vec A-\vec B\vert ^2=4\vec A\cdot\vec B=0\Rightarrow \vec A\cdot\vec B=0$.

Example 5.A2. Given $\vec A$, $\vec B$ with $A=B=1$ and $\vert \vec A\times\vec B\vert ^2 + \vert \vec A\cdot\vec B\vert ^2$ equal to?

$\vert A\times B\vert ^2=A^2B^2\sin^2\theta$; $\vert A\cdot B\vert ^2=A^2B^2\cos^2\theta$. Sum $=A^2B^2=1$.

This identity, $\vert \vec A\times\vec B\vert ^2 + (\vec A\cdot\vec B)^2 = A^2 B^2$, is called the Lagrange identity and frequently appears in JEE Advanced as a way to find an unknown angle.

Example 5.A3. Three vectors $\vec a, \vec b, \vec c$ form a closed triangle, $\vec a+\vec b+\vec c=\vec 0$. Show $\vec a\times\vec b=\vec b\times\vec c=\vec c\times\vec a$.

From $\vec c=-\vec a-\vec b$: $\vec b\times\vec c = \vec b\times(-\vec a-\vec b) = -\vec b\times\vec a -\vec 0=\vec a\times\vec b$. Similarly $\vec c\times\vec a=\vec a\times\vec b$. Geometrically the three give twice the area of the triangle.

Problem-Solving Heuristics

1. Always write the dimensional formula in $[M^a L^b T^c]$ form — never carry units along symbols.
2. In error problems, convert everything to percentages before adding; this avoids unit clutter.
3. Use logarithmic differentiation for any product/quotient/power; never use repeated chain rule.
4. Resolve along convenient axes — choose axes such that maximum number of forces/vectors lie along them.
5. To find an angle between vectors, compute both $\vec A\cdot\vec B$ and $\vert \vec A\times\vec B\vert$; their ratio is $\cot\theta$ — gives $\theta$ without ambiguity in $[0,\pi]$.
6. If the question says "minimum number of vectors", remember three coplanar non-collinear vectors of any magnitudes can sum to zero, but you need at least four non-coplanar vectors in 3D.
7. Dimensional analysis can give the dependence but never the dimensionless constant — JEE Advanced often hides $\pi$, $2\pi$, $1/2$.

Common Traps & Mistakes

• Confusing work ($\vec F\cdot\vec d$, scalar) with torque ($\vec r\times\vec F$, vector) — both have units of $\text{N·m}$ and dimension $ML^2T^{-2}$, but they are entirely different physical objects.
• Trigonometric functions require dimensionless arguments. If $y = A\sin(\omega t+\phi)$, both $\omega t$ and $\phi$ are dimensionless.
• In error analysis, when a quantity occurs in both numerator and denominator with the same power, errors do NOT cancel in the worst-case (maximum) formula — they add. They only cancel in the most-probable (RSS) formula.
• $\Delta x$ vs $dx$: $dx$ is infinitesimal; $\Delta x$ is finite uncertainty.
• $\vec A\cdot\vec B=0$ implies $\vec A\perp\vec B$ only if $\vec A,\vec B\neq\vec 0$.
• $\hat\imath\times\hat\jmath=\hat k$, not $-\hat k$. Many students reverse this on the spot in panic.
• A vector with zero magnitude is the null vector $\vec 0$; it has arbitrary direction.

Quick Revision Card

• 7 base SI units; supplementary radians & steradians.
• Homogeneity: every term in a physical equation has same dimensions.
• Conversion: $n_2 = n_1 (M_1/M_2)^a(L_1/L_2)^b(T_1/T_2)^c$.
• Error in $X = A^p B^q C^r$: $\Delta X/X = \vert p\vert \Delta A/A + \vert q\vert \Delta B/B + \vert r\vert \Delta C/C$.
• $R=\sqrt{A^2+B^2+2AB\cos\theta}$, $\tan\alpha=B\sin\theta/(A+B\cos\theta)$.
• Lagrange identity: $\vert \vec A\times\vec B\vert ^2+(\vec A\cdot\vec B)^2=A^2B^2$.
• BAC-CAB: $\vec A\times(\vec B\times\vec C)=(\vec A\cdot\vec C)\vec B-(\vec A\cdot\vec B)\vec C$.

Formula Sheet