Download App

The plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsThe plane5 Marks
Question
Find the distance between the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})+7=0$ and $\vec{\text{r}}\cdot(2\hat{\text{i}}+4\hat{\text{j}}+6\hat{\text{k}})+7=0$
✓

Answer

The given plane are,
$\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=-7$
$\Rightarrow\text{x}+2\text{y}+3\text{z}=-7$
Multiplying this equation of the plane by 2, we get
$2​​\text{x}+4​\text{y}+6​\text{z}=-14\ ...(\text{i})$
and
$\vec{\text{r}}\cdot(2\hat{\text{i}}+4\hat{\text{j}}+6\hat{\text{k}})=-7$
$2​​\text{x}+4​\text{y}+6​\text{z}=-7\ ...(\text{ii})$
We know that distance between two planes $ax + by + cz = d_1 $ and $ax + by + cz = d_2$ is $\frac{\big|\text{d}_2-\text{d}_1\big|}{\sqrt{\text{a}^2+\text{b}^2+\text{c}^2}}$
So, the required distance
$=\frac{|-7-(-14)|}{\sqrt{2^2+4^2+6^2}}$
$=\frac{|7|}{\sqrt{4+16+36}}$
$=\frac{7}{\sqrt{56}}\text{ units}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from The plane→The plane — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following intregals:
$\int\frac{\text{x}^2}{(\text{x}-1)(\text{x}-2)(\text{x}-3)}\ \text{dx}$
→
Find the equation of a normal to the curve $y = x log_e $x which is parallel to the line $2x − 2y + 3 = 0.$
→
Show that the function $\text{f(x)}\begin{cases}\text{x}^\text{m}\sin(\frac{1}{\text{x}}), &\text{x}\neq0 \\0 ,& \text{x}=0\end{cases}$
Continuous but not diffierentiable at x = 0, if 0 < m < 1
→
Evaluate : $\int \frac{2 x-3}{3 x^2+4 x+5} \cdot d x$
→
Differentiate the following w.r.t. x:

$\left[(\tan x)^{\tan x}\right]^{\tan x}$ at $x=\frac{\pi}{4}$

→
Evaluate the following definite integrals:$\int_{-\frac{\pi}{4}}^\limits{\frac{\pi}{4}}\frac{1}{1+\sin\text{x}}\text{ dx}$
→
If $\text{A}=\begin{bmatrix}\cos\theta&\text{i}\sin\theta\\\text{i}\sin\theta&\cos\theta\end{bmatrix},$ then prove by principle of mathematical induction that $\text{A}^\text{n}=\begin{bmatrix}\cos\text{n}\theta&\text{i}\sin\text{n}\theta\\\text{i}\sin\text{n}\theta&\cos\text{n}\theta\end{bmatrix}$ for all $\text{n}\in\text{N}.$
→
$\int\frac{\text{x}^2+5\text{x}+2}{\text{x}+2}\text{dx}$
→
Find the condition for the following set of curves to intersect orthogonally
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\frac{\text{x}^2}{\text{A}^2}-\frac{\text{y}^2}{\text{B}^2}=1$
→
Evaluvate the following intregals:
$\int\frac{5\text{x}+3}{\sqrt{\text{x}^2+4\text{x}+10}}\ \text{dx}$
→