Let the line L_1$L_{1}$ be parallel to the vector -3hati + 2hatj + 4hatk$-3\hat{i} + 2\hat{j} + 4\hat{k}$ and pass through the point (2, 6, 7) and the line L_2$L_{2}$ be parallel to the vector 2hati + hatj + 3hatk$2\hat{i} + \hat{j} + 3\hat{k}$ and pass through the point (4, 3, 5). If the line L_3$L_{3}$ is parallel to the vector -3hati + 5hatj + 16hatk$-3\hat{i} + 5\hat{j} + 16\hat{k}$ and intersects the lines L_1$L_{1}$ and L_2$L_{2}$ at the points C and D, respectively, then left|overrightarrowCDright|^2$\left|\overrightarrow{CD}\right|^{2}$ is equal to:

### Related Formula textEquation of a line: vecr = veca + lambdavecb$$\text{Equation of a line: } \vec{r} = \vec{a} + \lambda\vec{b}$$ |overrightarrowCD|^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2$$|\overrightarrow{CD}|^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2$$ ### Core Logic Write general points on L_1$L_1$ and L_2$L_2$ representing points C$C$ and D$D$. The vector overrightarrowCD$\overrightarrow{CD}$ must be parallel to the given direction vector of L_3$L_3$. This yields proportional equations to solve for the line parameters. ### Step 1: Write Line Equations Line L_1$L_{1}$: fracx-2-3 = fracy-62 = fracz-74 = lambda_1$\frac{x-2}{-3} = \frac{y-6}{2} = \frac{z-7}{4} = \lambda_{1}$ Point C$C$ on L_1$L_{1}$: (-3lambda_1+2, 2lambda_1+6, 4lambda_1+7)$(-3\lambda_{1}+2, 2\lambda_{1}+6, 4\lambda_{1}+7)$ Line L_2$L_{2}$: fracx-42 = fracy-31 = fracz-53 = lambda_2$\frac{x-4}{2} = \frac{y-3}{1} = \frac{z-5}{3} = \lambda_{2}$ Point D$D$ on L_2$L_{2}$: (2lambda_2+4, lambda_2+3, 3lambda_2+5)$(2\lambda_{2}+4, \lambda_{2}+3, 3\lambda_{2}+5)$ ### Step 2: Proportionality of Vector CD The vector overrightarrowCD = langle 2lambda_2+3lambda_1+2, lambda_2-2lambda_1-3, 3lambda_2-4lambda_1-2 rangle$\overrightarrow{CD} = \langle 2\lambda_{2}+3\lambda_{1}+2, \lambda_{2}-2\lambda_{1}-3, 3\lambda_{2}-4\lambda_{1}-2 \rangle$. Since L_3$L_3$ is parallel to -3hati + 5hatj + 16hatk$-3\hat{i} + 5\hat{j} + 16\hat{k}$, the components are proportional: frac2lambda_2+3lambda_1+2-3 = fraclambda_2-2lambda_1-35 = frac3lambda_2-4lambda_1-216$$\frac{2\lambda_{2}+3\lambda_{1}+2}{-3} = \frac{\lambda_{2}-2\lambda_{1}-3}{5} = \frac{3\lambda_{2}-4\lambda_{1}-2}{16}$$ ### Step 3: Solve for lambda values From the first two expressions: 5(2lambda_2+3lambda_1+2) = -3(lambda_2-2lambda_1-3)$$5(2\lambda_{2}+3\lambda_{1}+2) = -3(\lambda_{2}-2\lambda_{1}-3)$$ 10lambda_2 + 15lambda_1 + 10 = -3lambda_2 + 6lambda_1 + 9$$10\lambda_{2} + 15\lambda_{1} + 10 = -3\lambda_{2} + 6\lambda_{1} + 9$$ 13lambda_2 + 9lambda_1 = -1$$13\lambda_{2} + 9\lambda_{1} = -1$$ From the last two expressions: 16(lambda_2-2lambda_1-3) = 5(3lambda_2-4lambda_1-2)$$16(\lambda_{2}-2\lambda_{1}-3) = 5(3\lambda_{2}-4\lambda_{1}-2)$$ 16lambda_2 - 32lambda_1 - 48 = 15lambda_2 - 20lambda_1 - 10$$16\lambda_{2} - 32\lambda_{1} - 48 = 15\lambda_{2} - 20\lambda_{1} - 10$$ lambda_2 - 12lambda_1 = 38$$\lambda_{2} - 12\lambda_{1} = 38$$ Substitute lambda_2 = 12lambda_1 + 38$\lambda_{2} = 12\lambda_{1} + 38$ into the first equation: 13(12lambda_1 + 38) + 9lambda_1 = -1$$13(12\lambda_{1} + 38) + 9\lambda_{1} = -1$$ 156lambda_1 + 494 + 9lambda_1 = -1$$156\lambda_{1} + 494 + 9\lambda_{1} = -1$$ 165lambda_1 = -495 implies lambda_1 = -3$$165\lambda_{1} = -495 \implies \lambda_{1} = -3$$ lambda_2 = 12(-3) + 38 = 2$$\lambda_{2} = 12(-3) + 38 = 2$$ Coordinates of C$C$: (11, 0, -5)$(11, 0, -5)$ Coordinates of D$D$: (8, 5, 11)$(8, 5, 11)$ ### Step 4: Calculate Magnitude squared left|overrightarrowCDright|^2 = (8 - 11)^2 + (5 - 0)^2 + (11 - (-5))^2$$\left|\overrightarrow{CD}\right|^{2} = (8 - 11)^2 + (5 - 0)^2 + (11 - (-5))^2$$ = (-3)^2 + 5^2 + 16^2$$= (-3)^2 + 5^2 + 16^2$$ = 9 + 25 + 256 = 290$$= 9 + 25 + 256 = 290$$ ### Pattern Recognition For intersecting lines via a transversal of known direction, represent intersection points generally using independent parameters lambda$\lambda$ and mu$\mu$. The difference vector MUST be proportional to the given direction ratio. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Three Dimensional Geometry Class 12 Maths: Vector Algebra