A10.0
θ37°
Ax8.0
Ay6.0
ⓘ Change the direction and notice how Ax and Ay redistribute.
A question to think about
A car travels 20 km.
Do we know where it reached?
Not yet. We only know that the car travelled 20 km. It could have moved in any direction from point A. To locate point B, we also need to know the direction from A to B. That is why displacement is a vector.
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Meet vectors
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Scalars & vectors
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Anatomy of a vector
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Position & displacement vectors
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Types of vectors
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Read vectors
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Coordinate axes & unit vectors
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Resolving a vector into components
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Magnitude from components
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Direction from components
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Unit vector along a direction
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Vector between two coordinates
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Operate on vectors
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Multiplication by a scalar
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Vector addition — graphical methods
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Vector addition — analytical method
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Resultant limits & equilibrium
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Vector subtraction
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Dot product & projection
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Cross product
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Go beyond
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Triple products & vector identities
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Practise and solve
1UnderstandRead the situation
2DrawShow the vectors
3ResolveFind components
4OperateChoose the method
5CheckMagnitude & direction
Build your skills
3questions
Start practice →
10 ladders · Start with Practice 1.1
Chapter 3 · Vectors
Solve through experimentation
2 visual challenges available Open visual challenges →Choose a question
Linked questions will appear when published.Can you connect the ideas?
2 mixed questions · Recommended after practice Start checkpoint →Master
Think beyond the textbook
NEET Sprint
Fast conceptual decisions, sign awareness and accurate substitution.
Start practice →JEE Main Challenge
Multi-step vector problems solved with speed, structure and precision.
Start practice →JEE Advanced Thinking
Connect identities, geometry and unfamiliar constraints.
Start challenge →Revise & remember
Make the idea stick
Formula map
ƒMeaning before memory
Every relationship is grouped by the idea it describes.
Meet\(\vec A=A_x\hat i+A_y\hat j+A_z\hat k\)Understand vector meaning
Read\(|\vec A|=\sqrt{A_x^2+A_y^2+A_z^2}\)Read magnitude and direction
Operate\(\vec R=k\vec A,\quad|\vec R|=|k|\,|\vec A|\)Combine vector effects
Go beyond\(\vec A\cdot(\vec B\times\vec C)=\det[\vec A\;\vec B\;\vec C]\)Connect advanced ideas
Meet vectors
Vector representation\(\vec A=A_x\hat i+A_y\hat j+A_z\hat k\) Position vector\(\vec r=x\hat i+y\hat j+z\hat k\) Displacement vector\(\overrightarrow{AB}=\vec r_B-\vec r_A\) Negative vector\(-\vec A\)Read vectors
Magnitude of a Vector\(|\vec A|=\sqrt{A_x^2+A_y^2+A_z^2}\) Unit vector along A\(\hat A=\frac{\vec A}{|\vec A|}\) Rectangular components\(A_x=A\cos\theta,\quad A_y=A\sin\theta\) Direction from components\(\theta=\tan^{-1}\!\left(\frac{A_y}{A_x}\right)\) Direction cosines\(\cos\alpha=\frac{A_x}{A},\quad\cos\beta=\frac{A_y}{A},\quad\cos\gamma=\frac{A_z}{A}\) Direction-cosine identity\(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\)Operate on vectors
Scalar multiplication\(\vec R=k\vec A,\quad|\vec R|=|k|\,|\vec A|\) Vector addition and subtraction\(\vec R=\vec A\pm\vec B,\quad R_x=A_x\pm B_x\) Vector Addition\(R=\sqrt{A^2+B^2+2AB\cos\theta}\) Resultant direction\(\tan\phi=\frac{B\sin\theta}{A+B\cos\theta}\) Dot product\(\vec A\cdot\vec B=A_xB_x+A_yB_y+A_zB_z=AB\cos\theta\) Vector projection\(\operatorname{proj}_{\vec B}\vec A=\frac{\vec A\cdot\vec B}{|\vec B|^2}\vec B\) Cross-product magnitude\(|\vec A\times\vec B|=AB\sin\theta\) Areas from cross product\(A_{\mathrm{parallelogram}}=|\vec A\times\vec B|,\quad A_{\triangle}=\frac12|\vec A\times\vec B|\)Go beyond
Scalar triple product\(\vec A\cdot(\vec B\times\vec C)=\det[\vec A\;\vec B\;\vec C]\) Volumes from triple product\(V_{\mathrm{parallelepiped}}=|\vec A\cdot(\vec B\times\vec C)|,\quad V_{\mathrm{tetrahedron}}=\frac16|\vec A\cdot(\vec B\times\vec C)|\) Coplanarity condition\(\vec A\cdot(\vec B\times\vec C)=0\) Vector triple product\(\vec A\times(\vec B\times\vec C)=\vec B(\vec A\cdot\vec C)-\vec C(\vec A\cdot\vec B)\) Lagrange identity\(|\vec A\times\vec B|^2=A^2B^2-(\vec A\cdot\vec B)^2\)Concept journey
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