Download App

The plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsThe plane3 Marks
Question
Find the angle between the given planes.
$\vec{\text{r}}\cdot(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}})=1$ and $\vec{\text{r}}\cdot(-\hat{\text{i}}+\hat{\text{j}})=4$
✓

Answer

We know that the angle between the planes $\vec{\text{r}}\cdot\vec{\text{n}}_1=\text{d}_1,\vec{\text{ r}}\cdot\vec{\text{n}}_2=\text{d}_2$ is given by
$\cos\theta=\frac{\vec{\text{n}}_1\cdot\vec{\text{n}}_2}{|\vec{\text{n}}_1||\vec{\text{n}}_2|}$
Here, $\vec{\text{n}}_1=2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$
$\vec{\text{n}}_2=-\hat{\text{i}}+\hat{\text{j}}+0\hat{\text{k}}$
So, $\cos\theta=\frac{(2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}})\cdot(-\hat{\text{i}}+\vec{\text{j}}+0\hat{\text{k}})}{\big|2\hat{\text{i}}-3\hat{\text{j}}+4{\hat{\text{k}}\big|}\big|\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}\big|}$
$=\frac{-2-3}{\sqrt{4+9+16}\sqrt{1+1+0}}$
$=\frac{-5}{\sqrt{29}\sqrt{2}}$
$=\frac{-5}{\sqrt{58}}$
$\theta=\cos^{-1}\Big(\frac{-5}{\sqrt{58}}\Big)$

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.(3 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

Find the vector equation of the line passing through the point having position vector

$\hat{i}+2 \hat{j}+3 \hat{k}$ and perpendicular to vectors $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}-\hat{j}+\hat{k}$.

→
Find the principal solutions of the following equations : $\sin 2 \theta=-\frac{1}{2}$
→
Let $f : R → R$ be defined as $\text{f(x)}=\frac{2\text{x}-3}{4}.$ Write $fof ^{-1}(1)$.
→
Find the distance of the point $\hat{i}+2 \hat{\jmath}-\hat{k}$ from the plane $\bar{r} \cdot(\hat{i}-2 \hat{j}+4 \hat{k})=10$.
→
The profit function $P(x)$ of a firm, selling x items per day is given by $P(x) = (150 – x)x – 1625.$ Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
→
Evaluate the following :

$\int \sqrt{\frac{9+x}{9-x}} \cdot d x$

→
If $A =\left[\begin{array}{ll}4 & 5 \\ 2 & 1\end{array}\right]$, show that $A ^{-1}=\frac{1}{6}( A -5 I )$.
→
Find the direction cosines and direction angles of the vector.

$(1 / 2) \hat{i}+\hat{j}+\hat{k}$

→
Differentiate the following functions with respect to x:
$\text{e}^{\sin^{-1}2\text{x}}$
→
In $\triangle A B C$ prove that $a^2\left(\cos ^2 B-\cos ^2 C\right)+b^2\left(\cos ^2 C-\cos ^2 A\right)+c^2\left(\cos ^2 A-\cos ^2 B\right)=0$.
→