Download App

The plane — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThe plane4 Marks
Question
Find the vector equation of the following planes in non-parametric form.
$\vec{\text{r}}=(\lambda-2\mu)\hat{\text{i}}+(3-\mu)\hat{\text{j}}+(2\lambda+\mu)\hat{\text{k}}$
✓

Answer

The given equation of the plane is,
$\vec{\text{r}}=(\lambda-2\mu)\hat{\text{i}}+(3-\mu)\hat{\text{j}}+(2\lambda+\mu)\hat{\text{k}}$
$\Rightarrow\vec{\text{r}}=(0\hat{\text{i}}+3\hat{\text{j}}+0\hat{\text{k}})+\lambda(\hat{\text{i}}+0\hat{\text{j}}+2\hat{\text{k}})+\mu(-2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})$
We know that equation $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}+\mu\vec{\text{c}} $ represents a plane passing through a point whose position vector is $\vec{\text{a}}$ and parallel to t.
Here, $\vec{\text{a}}=0\hat{\text{i}}+3\hat{\text{j}}+0\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+0\hat{\text{j}}+2\hat{\text{k}},\hat{\text{c}}=-2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
Normal vector, $\vec{\text{n}}=\vec{\text{b}}\times\vec{\text{c}}$
$=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\1&0&2\\-2&-1&1\end{vmatrix}$
$=2\hat{\text{i}}-5\hat{\text{j}}-\hat{\text{k}}$
The vector equation of the plane in scalar product form is,
$\vec{\text{r}}\cdot\vec{\text{n}}=\vec{\text{a}}\cdot\vec{\text{n}}$
$\Rightarrow\vec{\text{r}}\cdot(2\hat{\text{i}}-5\hat{\text{j}}-\hat{\text{k}})=(0\hat{\text{i}}+3\hat{\text{j}}+0\hat{\text{k}})(2\hat{\text{i}}-5\hat{\text{j}}-\hat{\text{k}})$
$\Rightarrow\vec{\text{r}}\cdot(2\hat{\text{i}}-5\hat{\text{j}}-\hat{\text{k}})=0-15+0$
$\Rightarrow\vec{\text{r}}\cdot(2\hat{\text{i}}-5\hat{\text{j}}-\hat{\text{k}})+15=0$

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 "4 Marks" from The plane→The plane — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that $\text{f}(\text{x})=\text{x}^\frac{1}{3}$ is not differentible at x = 0.
→
Maximum Z = 3x + 4y

Subject to

$\text{x}+\text{y}\leq30000$

$\text{y}\leq12000$

$\text{x}\geq6000$

$\text{x}\geq\text{y}$

$\text{x},\text{y}\geq0$

→
Evaluate the following integrals:
$\int\limits^{{\pi}}_{{-\pi}}\frac{2\text{x}(1+\sin\text{x})}{1+\cos^2\text{x}}\text{ dx}$
→
If y = tan-1 $\Bigg[\frac{\sqrt{1+\text{x}^{2}}-\sqrt{1-\text{x}^{2}}}{\sqrt{1+\text{x}^{2}}+\sqrt{1-\text{x}^{2}}}\Bigg],\text{ find }\frac{\text{dy}}{\text{dx}}.$
→
If $\text{y}=\tan^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big),$, find $\frac{\text{dy}}{\text{dx}}.$
→
 Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$ and
$\text{G}(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$
Show that
$\big[\text{F}(\alpha)\big]^{-1}=\text{F}(-\alpha)$
→
If $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ 2\hat{\text{i}}+5\hat{\text{J}},\ 3\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}-6\hat{\text{j}}-\hat{\text{k}}$ respectively are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether $\vec{\text{AB}}$ and $\vec{\text{CD}}$ are collinear or not.
→
Prove that:
$\begin{vmatrix}1&\text{a}^2+\text{bc}&\text{a}^3\\1&\text{b}^2+\text{ca}&\text{b}^3\\1&\text{c}^2+\text{ab}&\text{c}^3 \end{vmatrix}$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}^2+\text{b}^2+\text{c}^2)$
→
Prove that function $\sin ^2 x(1+\cos x)$, at $\cos x$ $=\frac{1}{3}$ is maximum.
→
Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i} + 3\hat{j} + 4\hat{k}$ to the plane $\overrightarrow{\text{r}}. (2\hat{i} + \hat{j} + 3\hat{k}) - 26 = 0.$ Also find image of P in the plane.
→