Vectors & 3D Geometry (JEE)

Cheatsheet Content

1. Vectors Basics Vector: Quantity with magnitude and direction. Represented as $\vec{a}$ or $\mathbf{a}$. Position Vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. Magnitude $|\vec{r}| = \sqrt{x^2+y^2+z^2}$. Unit Vector: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$. Direction cosines: $\cos\alpha = \frac{x}{|\vec{r}|}$, $\cos\beta = \frac{y}{|\vec{r}|}$, $\cos\gamma = \frac{z}{|\vec{r}|}$. $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$ $\hat{a} = (\cos\alpha)\hat{i} + (\cos\beta)\hat{j} + (\cos\gamma)\hat{k}$ Vector Addition: $\vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}$ Section Formula: Internal: $\frac{m\vec{b} + n\vec{a}}{m+n}$ External: $\frac{m\vec{b} - n\vec{a}}{m-n}$ 2. Dot Product (Scalar Product) Definition: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ In components: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$ Properties: Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$ $\vec{a} \cdot \vec{a} = |\vec{a}|^2$ $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$ Angle between vectors: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$ Orthogonality: If $\vec{a} \perp \vec{b}$, then $\vec{a} \cdot \vec{b} = 0$. Projection: Projection of $\vec{a}$ on $\vec{b}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$. Vector projection is $\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$. 3. Cross Product (Vector Product) Definition: $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$, where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$. In components: $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$ Properties: Anti-commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$ Distributive: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$ $\vec{a} \times \vec{a} = \vec{0}$ $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$ Area of Parallelogram: $|\vec{a} \times \vec{b}|$ Area of Triangle: $\frac{1}{2}|\vec{a} \times \vec{b}|$ Collinearity: If $\vec{a} \parallel \vec{b}$, then $\vec{a} \times \vec{b} = \vec{0}$. 4. Scalar Triple Product (Box Product) Definition: $[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$ In components: $[\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$ Properties: Cyclic permutation: $[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}]$ Order change: $[\vec{a} \vec{b} \vec{c}] = -[\vec{b} \vec{a} \vec{c}]$ If any two vectors are identical or parallel, STP is 0. Volume of Parallelepiped: $|[\vec{a} \vec{b} \vec{c}]|$ Volume of Tetrahedron: $\frac{1}{6}|[\vec{a} \vec{b} \vec{c}]|$ Coplanarity: If $\vec{a}, \vec{b}, \vec{c}$ are coplanar, then $[\vec{a} \vec{b} \vec{c}] = 0$. 5. Vector Triple Product Definition: $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$ (BAC - CAB rule) Note: $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a}$ 6. 3D Coordinate System Distance Formula: Between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. Section Formula: Internal: $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n}\right)$ External: $\left(\frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}, \frac{mz_2-nz_1}{m-n}\right)$ Centroid of Triangle: $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$ Centroid of Tetrahedron: $\left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4}\right)$ 7. Direction Cosines & Ratios Direction Cosines (DC's): $\ell = \cos\alpha, m = \cos\beta, n = \cos\gamma$. $\ell^2+m^2+n^2=1$. Direction Ratios (DR's): Any set $a, b, c$ proportional to DC's. $\frac{\ell}{a} = \frac{m}{b} = \frac{n}{c} = \frac{1}{\sqrt{a^2+b^2+c^2}}$. DC's of line joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $\frac{x_2-x_1}{d}, \frac{y_2-y_1}{d}, \frac{z_2-z_1}{d}$ where $d$ is the distance. Angle between two lines: $\cos\theta = \ell_1\ell_2 + m_1m_2 + n_1n_2$ (using DC's). Or $\cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$ (using DR's). 8. Straight Line in 3D Equation of line through $(x_1, y_1, z_1)$ with DR's $a, b, c$: Cartesian: $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$ Vector: $\vec{r} = \vec{a} + \lambda\vec{b}$, where $\vec{a}$ is position vector of point, $\vec{b}$ is direction vector. Equation of line through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: Cartesian: $\frac{x-x_1}{x_2-x_1} = \frac{y-y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1}$ Vector: $\vec{r} = \vec{a} + \lambda(\vec{b}-\vec{a})$ Angle between two lines: $\cos\theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}||\vec{b_2}|}$ (vector form). Shortest Distance between Skew Lines: Lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ $d = \left|\frac{(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})}{|\vec{b_1} \times \vec{b_2}|}\right|$ Shortest Distance between Parallel Lines: Lines $\vec{r} = \vec{a_1} + \lambda\vec{b}$ and $\vec{r} = \vec{a_2} + \mu\vec{b}$ $d = \left|\frac{(\vec{a_2} - \vec{a_1}) \times \vec{b}}{|\vec{b}|}\right|$ 9. Plane in 3D Equation of Plane: Normal Form: $\vec{r} \cdot \hat{n} = d$, where $\hat{n}$ is unit normal and $d$ is distance from origin. Cartesian Normal Form: $\ell x + my + nz = d$ General Form: $Ax+By+Cz+D=0$. Normal vector $\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$. Through a point $(x_1, y_1, z_1)$ and normal $\vec{n} = A\hat{i} + B\hat{j} + C\hat{k}$: $A(x-x_1) + B(y-y_1) + C(z-z_1) = 0$. Through three non-collinear points $(x_1,y_1,z_1)$, $(x_2,y_2,z_2)$, $(x_3,y_3,z_3)$: $\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$. Intercept Form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ (intercepts $a, b, c$ on axes). Angle between two planes: $\cos\theta = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}||\vec{n_2}|}$. (Angle between their normals). Angle between a line and a plane: $\sin\theta = \frac{\vec{b} \cdot \vec{n}}{|\vec{b}||\vec{n}|}$. (Angle between line direction vector $\vec{b}$ and plane normal $\vec{n}$). Distance of a point $(x_1, y_1, z_1)$ from a plane $Ax+By+Cz+D=0$: $d = \frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$. Distance between two parallel planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$: $d = \frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}$. Equation of plane passing through intersection of two planes $P_1=0$ and $P_2=0$: $P_1 + \lambda P_2 = 0$. $(A_1x+B_1y+C_1z+D_1) + \lambda(A_2x+B_2y+C_2z+D_2) = 0$ Coplanarity of four points: $(x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3), (x_4,y_4,z_4)$ are coplanar if $\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \\ x_4-x_1 & y_4-y_1 & z_4-z_1 \end{vmatrix} = 0$.