Lines and Planes Class 12

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1. Vector Algebra Basics Position Vector: For a point $P(x, y, z)$, its position vector is $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. Vector from point $A$ to $B$: If $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, then $\vec{AB} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$. Magnitude of a vector: If $\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}$, then $|\vec{a}| = \sqrt{x^2 + y^2 + z^2}$. Unit vector: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$. Scalar (Dot) Product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$. If $\vec{a} \perp \vec{b}$, then $\vec{a} \cdot \vec{b} = 0$. Vector (Cross) Product: $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}$. $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$. If $\vec{a} \parallel \vec{b}$, then $\vec{a} \times \vec{b} = \vec{0}$. 2. Equation of a Line 2.1. Vector Form Passing through a point $A(\vec{a})$ and parallel to vector $\vec{b}$: $\vec{r} = \vec{a} + \lambda\vec{b}$ where $\lambda$ is a scalar parameter. Passing through two points $A(\vec{a})$ and $B(\vec{b})$: $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$ 2.2. Cartesian Form Passing through a point $(x_1, y_1, z_1)$ and having direction ratios $(a, b, c)$: $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$ Passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$: $\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$ 3. Angle Between Two Lines Vector Form: If lines are $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$, $\cos\theta = \left| \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1||\vec{b}_2|} \right|$ Cartesian Form: If lines have direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$, $\cos\theta = \left| \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\sqrt{a_2^2 + b_2^2 + c_2^2}} \right|$ Condition for Perpendicular Lines: $\vec{b}_1 \cdot \vec{b}_2 = 0$ or $a_1a_2 + b_1b_2 + c_1c_2 = 0$. Condition for Parallel Lines: $\vec{b}_1 = k\vec{b}_2$ or $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. 4. Shortest Distance Between Two Skew Lines Vector Form: Lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$. $d = \left| \frac{(\vec{b}_1 \times \vec{b}_2) \cdot (\vec{a}_2 - \vec{a}_1)}{|\vec{b}_1 \times \vec{b}_2|} \right|$ Shortest distance between parallel lines: $\vec{r} = \vec{a}_1 + \lambda\vec{b}$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}$. $d = \left| \frac{\vec{b} \times (\vec{a}_2 - \vec{a}_1)}{|\vec{b}|} \right|$ 5. Equation of a Plane 5.1. Normal Form Vector Form: $\vec{r} \cdot \hat{n} = d$ where $\hat{n}$ is the unit normal vector from the origin to the plane, and $d$ is the perpendicular distance from the origin to the plane. Cartesian Form: $lx + my + nz = d$ where $l, m, n$ are direction cosines of the normal to the plane. 5.2. General Form (Passing through a point and perpendicular to a vector) Vector Form: Passing through $A(\vec{a})$ and normal to $\vec{N}$: $(\vec{r} - \vec{a}) \cdot \vec{N} = 0 \quad \text{or} \quad \vec{r} \cdot \vec{N} = \vec{a} \cdot \vec{N}$ Cartesian Form: Passing through $(x_1, y_1, z_1)$ and normal to $A\hat{i} + B\hat{j} + C\hat{k}$: $A(x - x_1) + B(y - y_1) + C(z - z_1) = 0$ General Cartesian Equation: $Ax + By + Cz + D = 0$ where $A, B, C$ are direction ratios of the normal to the plane. 5.3. Intercept Form Cartesian Form: Cutting intercepts $a, b, c$ on the axes: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ 5.4. Passing Through Three Non-Collinear Points Vector Form: Passing through $A(\vec{a}), B(\vec{b}), C(\vec{c})$. $(\vec{r} - \vec{a}) \cdot [(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})] = 0$ Cartesian Form: For points $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$. $\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$ 5.5. Plane Passing Through the Intersection of Two Planes Vector Form: For planes $\vec{r} \cdot \vec{N}_1 = d_1$ and $\vec{r} \cdot \vec{N}_2 = d_2$: $\vec{r} \cdot (\vec{N}_1 + \lambda\vec{N}_2) = d_1 + \lambda d_2$ Cartesian Form: For planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$: $(A_1x + B_1y + C_1z + D_1) + \lambda(A_2x + B_2y + C_2z + D_2) = 0$ 6. Angle Between Planes Vector Form: Between planes $\vec{r} \cdot \vec{N}_1 = d_1$ and $\vec{r} \cdot \vec{N}_2 = d_2$. $\cos\theta = \left| \frac{\vec{N}_1 \cdot \vec{N}_2}{|\vec{N}_1||\vec{N}_2|} \right|$ Cartesian Form: Between planes $A_1x + B_1y + C_1z + D_1 = 0$ and $A_2x + B_2y + C_2z + D_2 = 0$. $\cos\theta = \left| \frac{A_1A_2 + B_1B_2 + C_1C_2}{\sqrt{A_1^2 + B_1^2 + C_1^2}\sqrt{A_2^2 + B_2^2 + C_2^2}} \right|$ Condition for Parallel Planes: $\vec{N}_1 = k\vec{N}_2$ or $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$. Condition for Perpendicular Planes: $\vec{N}_1 \cdot \vec{N}_2 = 0$ or $A_1A_2 + B_1B_2 + C_1C_2 = 0$. 7. Angle Between a Line and a Plane Vector Form: Between line $\vec{r} = \vec{a} + \lambda\vec{b}$ and plane $\vec{r} \cdot \vec{N} = d$. $\sin\theta = \left| \frac{\vec{b} \cdot \vec{N}}{|\vec{b}||\vec{N}|} \right|$ Cartesian Form: Between line $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and plane $A_2x + B_2y + C_2z + D_2 = 0$. $\sin\theta = \left| \frac{A_2a_1 + B_2b_1 + C_2c_1}{\sqrt{A_2^2 + B_2^2 + C_2^2}\sqrt{a_1^2 + b_1^2 + c_1^2}} \right|$ Condition for Line Parallel to Plane: $\vec{b} \cdot \vec{N} = 0$ or $A_2a_1 + B_2b_1 + C_2c_1 = 0$. Condition for Line Perpendicular to Plane: $\vec{b} \parallel \vec{N}$ or $\frac{a_1}{A_2} = \frac{b_1}{B_2} = \frac{c_1}{C_2}$. 8. Distance of a Point from a Plane Vector Form: Distance of point $P(\vec{p})$ from plane $\vec{r} \cdot \vec{N} = d$. $D = \frac{|\vec{p} \cdot \vec{N} - d|}{|\vec{N}|}$ Cartesian Form: Distance of point $(x_1, y_1, z_1)$ from plane $Ax + By + Cz + D_0 = 0$. $D = \frac{|Ax_1 + By_1 + Cz_1 + D_0|}{\sqrt{A^2 + B^2 + C^2}}$ 9. Coplanarity of Two Lines Vector Form: Lines $\vec{r} = \vec{a}_1 + \lambda\vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu\vec{b}_2$ are coplanar if $(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0$ Cartesian Form: Lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$ are coplanar if $\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0$