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Class 11 Physics

Physics Formula Sheets

Quickly revise the important formulas from each chapter of Class 11 Physics.

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Chapter 3

Motion in a Plane

Meet vectors

Vector representation
\[\vec A=A_x\hat i+A_y\hat j+A_z\hat k\]

A vector written in Cartesian component form.

Position vector
\[\vec r=x\hat i+y\hat j+z\hat k\]

The position of a point measured from the chosen origin.

Displacement vector
\[\overrightarrow{AB}=\vec r_B-\vec r_A\]

Final position vector minus initial position vector.

Negative vector
\[-\vec A\]

A vector with the same magnitude as A and the opposite direction.

Read vectors

Magnitude of a Vector
\[|\vec A|=\sqrt{A_x^2+A_y^2+A_z^2}\]

Magnitude obtained from the Cartesian components.

Unit vector along A
\[\hat A=\frac{\vec A}{|\vec A|}\]

Divide a non-zero vector by its magnitude.

Rectangular components
\[A_x=A\cos\theta,\quad A_y=A\sin\theta\]

Components when theta is measured from the positive x-axis.

Direction from components
\[\theta=\tan^{-1}\!\left(\frac{A_y}{A_x}\right)\]

Use the signs of both components to place the direction in the correct quadrant.

Direction cosines
\[\cos\alpha=\frac{A_x}{A},\quad\cos\beta=\frac{A_y}{A},\quad\cos\gamma=\frac{A_z}{A}\]

Angles made by a vector with the positive coordinate axes.

Direction-cosine identity
\[\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\]

Identity obeyed by the three direction cosines.

Operate on vectors

Scalar multiplication
\[\vec R=k\vec A,\quad|\vec R|=|k|\,|\vec A|\]

A scalar changes a vector’s magnitude and may reverse its direction.

Vector addition and subtraction
\[\vec R=\vec A\pm\vec B,\quad R_x=A_x\pm B_x\]

Add or subtract corresponding components.

Vector Addition
\[R=\sqrt{A^2+B^2+2AB\cos\theta}\]

Magnitude of the resultant of two vectors separated by angle theta.

Resultant direction
\[\tan\phi=\frac{B\sin\theta}{A+B\cos\theta}\]

Direction of the resultant measured from vector A.

Dot product
\[\vec A\cdot\vec B=A_xB_x+A_yB_y+A_zB_z=AB\cos\theta\]

Scalar product in component and magnitude-angle forms.

Vector projection
\[\operatorname{proj}_{\vec B}\vec A=\frac{\vec A\cdot\vec B}{|\vec B|^2}\vec B\]

Vector projection of A along B.

Cross-product magnitude
\[|\vec A\times\vec B|=AB\sin\theta\]

Magnitude of the vector product.

Areas from cross product
\[A_{\mathrm{parallelogram}}=|\vec A\times\vec B|,\quad A_{\triangle}=\frac12|\vec A\times\vec B|\]

Areas spanned by two vectors.

Go beyond

Scalar triple product
\[\vec A\cdot(\vec B\times\vec C)=\det[\vec A\;\vec B\;\vec C]\]

Scalar triple product written as a determinant.

Volumes from triple product
\[V_{\mathrm{parallelepiped}}=|\vec A\cdot(\vec B\times\vec C)|,\quad V_{\mathrm{tetrahedron}}=\frac16|\vec A\cdot(\vec B\times\vec C)|\]

Volumes generated by three vectors.

Coplanarity condition
\[\vec A\cdot(\vec B\times\vec C)=0\]

Three vectors are coplanar when their scalar triple product is zero.

Vector triple product
\[\vec A\times(\vec B\times\vec C)=\vec B(\vec A\cdot\vec C)-\vec C(\vec A\cdot\vec B)\]

BAC–CAB identity.

Lagrange identity
\[|\vec A\times\vec B|^2=A^2B^2-(\vec A\cdot\vec B)^2\]

Relationship between dot and cross products.