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Vectors (Physics) revision notes

Every idea in the Vectors (Physics) chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 23 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Scalars, vectors and components

Scalars and vectors
  • A vector needs magnitude, direction and the triangle law of addition. Electric current has a direction but adds as a plain number, so it is a scalar; so are work, energy, power and pressure.
Magnitude and unit vector
5× asked, last 2025
∣A⃗∣=Ax2+Ay2+Az2,A^=A⃗∣A⃗∣|\vec A|=\sqrt{A_x^2+A_y^2+A_z^2},\qquad \hat A=\frac{\vec A}{|\vec A|}
A^\hat A has size 1 and no unit. i^,j^,k^\hat i,\hat j,\hat k point along +x+x, +y+y, +z+z.
  • Position vector r⃗=xi^+yj^+zk^\vec r=x\hat i+y\hat j+z\hat k. From point PP to point QQ: PQ⃗=r⃗Q−r⃗P\vec{PQ}=\vec r_Q-\vec r_P (final minus initial). This is the s⃗\vec s in W=F⃗⋅s⃗W=\vec F\cdot\vec s; unit vector along it: PQ⃗/∣PQ⃗∣\vec{PQ}/|\vec{PQ}|.
  • Quick method. A vector of size CC along B⃗\vec B is CB^C\hat B. Size of one vector, direction of another: find B^\hat B first, then scale it.
Components in a plane
5× asked, last 2024
Ax=Acos⁡θ,Ay=Asin⁡θ,tan⁡θ=AyAxA_x=A\cos\theta,\quad A_y=A\sin\theta,\qquad \tan\theta=\frac{A_y}{A_x}
θ\theta measured from the +x+x axis. The component along the axis the angle starts from takes cos⁡\cos.
xxyyAx=Acos⁡θA_x=A\cos\thetaAy=Asin⁡θA_y=A\sin\thetaA⃗\vec Aθ\theta
θ\theta from +x+x: the xx-part takes cos⁡\cos
xxyy225∘225^\circ−i^−j^-\hat i-\hat jnot 45∘45^\circ
tan⁡θ=1\tan\theta=1, both parts negative: 225∘225^\circ
  • tan⁡θ=Ay/Ax\tan\theta=A_y/A_x repeats every 180∘180^\circ: pick the quadrant from the signs of AxA_x and AyA_y (same for tan⁡ϕ=Ry/Rx\tan\phi=R_y/R_x). E.g. −i^−j^-\hat i-\hat j is at 225∘225^\circ, not 45∘45^\circ.
  • Not Ax=Acos⁡θA_x=A\cos\theta when θ\theta is from the yy-axis. Then Ay=Acos⁡θA_y=A\cos\theta and Ax=Asin⁡θA_x=A\sin\theta, because cosine goes with the axis the angle starts from.
  • Quick method. Many vectors at a point: add all xx-components, then all yy-components; R=Rx2+Ry2R=\sqrt{R_x^2+R_y^2}, tan⁡ϕ=Ry/Rx\tan\phi=R_y/R_x. A subtracted vector flips the signs of both its components.
Direction cosines
1× asked, last 2021
cos⁡α=AxA, cos⁡β=AyA, cos⁡γ=AzA,cos⁡2α+cos⁡2β+cos⁡2γ=1\cos\alpha=\frac{A_x}{A},\ \cos\beta=\frac{A_y}{A},\ \cos\gamma=\frac{A_z}{A},\qquad \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1
α,β,γ\alpha,\beta,\gamma = angles of A⃗\vec A with the xx, yy, zz axes. They are the components of A^\hat A.
xxyyzzA⃗\vec Aα\alphaβ\betaγ\gamma
cos⁡α,cos⁡β,cos⁡γ\cos\alpha,\cos\beta,\cos\gamma are the parts of A^\hat A along x,y,zx,y,z
  • The angle with the yy-axis can also be written tan⁡β=Ax2+Az2/Ay\tan\beta=\sqrt{A_x^2+A_z^2}/A_y. If the vector depends on time, put in tt first, then take components.
Resolving along two given lines
1× asked, last 2021
P1=Psin⁡βsin⁡(α+β),P2=Psin⁡αsin⁡(α+β)P_1=\frac{P\sin\beta}{\sin(\alpha+\beta)},\qquad P_2=\frac{P\sin\alpha}{\sin(\alpha+\beta)}
Line 1 at α\alpha, line 2 at β\beta from P⃗\vec P, on opposite sides. Lines at 90∘90^\circ: Pcos⁡αP\cos\alpha and Psin⁡αP\sin\alpha.
line 1line 2P1P_1P2P_2P⃗\vec Pα\alphaβ\beta
P⃗=P⃗1+P⃗2\vec P=\vec P_1+\vec P_2: a parallelogram, not two projections
  • Not Pcos⁡θP\cos\theta for the part along a rod when the two lines are not at 90∘90^\circ. The split is a parallelogram, not a projection. Use the triangle of forces: each part over the sine of the angle opposite it.

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Practise Vectors (Physics)

18 more concepts in 5 topics

Vector addition and subtraction, Equilibrium of concurrent forces, Dot product and Cross product, plus ideas from kinematics. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Vector addition and subtraction

  • Rules of vector algebra2×
  • Resultant of two vectors8×
  • Difference of two vectors9×
  • Two vectors of equal size5×
  • Triangle and polygon laws4×

Equilibrium of concurrent forces

  • Condition for equilibrium2×
  • Lami's theorem

Dot product

  • Dot (scalar) product10×
  • Perpendicular vectors6×
  • Component and projection3×
  • Work and power1×

Cross product

  • Cross (vector) product7×
  • Parallel vectors3×
  • Unit vector normal to a plane2×
  • Area from a cross product
  • Torque and angular momentum3×
  • Dot and cross together1×
  • Scalar triple product1×

Also asked here: kinematics

  • Key points1

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Practise Vectors (Physics)

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