Download App

The plane — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThe plane3 Marks
Question
The line $\vec{\text{r}}=\hat{\text{i}}+\lambda(2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}})$ is parallel to the plane $\vec{\text{r}}\cdot(\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}})=4.$ Find m.
✓

Answer

The given line is parallel to the vector $\vec{\text{b}}=2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}}$ and the given plane is normal to the vector $\vec{\text{n}}=\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}$
If the line is parallel to the plane, the normal to the plane is perpendicular to the line.
$\Rightarrow\vec{\text{b}}\perp\vec{\text{n}}$
$\Rightarrow\vec{\text{b}}\cdot\vec{\text{n}}=0$
$\Rightarrow(2\hat{\text{i}}-\text{m}\hat{\text{j}}-3\hat{\text{k}})\cdot(\text{m}\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}})=0$
$\Rightarrow2\text{m}-3\text{m}-3=0$
$\Rightarrow-\text{m}-3=0$
$\Rightarrow\text{m}=-3$

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 "3 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

Each of the following defines a relation on N:
x is greater than $\text{y},\ \text{x},\ \text{y}\in\text{N}$
Determine which of the above relations are reflexive, symmetric and transitive.
→
Three cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that the cards are a king, a queen and a jack.
→
Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2)(x) = f1(x) + f2(x) is not one-one.
→
If A, B, C, D are the points with position vectors $\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\ 2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}},\ 2\hat{\text{i}}-3\hat{\text{k}},$ respectively, find the projection of $\overrightarrow{\text{AB}}$ along $\overrightarrow{\text{CD}}.$
→
Find the angle between the lines
$
\begin{array}{ll}
& 2 x=3 y=-z \\
\text { and } & 6 x=-y=-4 z .
\end{array}
$
→
Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.
→
Find the local maxima and local minima of the function. Find also the local maximum and the local minimum value: f(x) = sinx - cosx, 0 < x < 2$ \pi$.
→
Prove that the relation R in the set $\text{A } = (1, 2, 3, 4, 5)$ given by $\text{R} = (\text{a, b)} : |\text{a-b|} \text{is even},$  is an equivalence relation.
→
Show that the lines $\frac{\text{x}-5}{7}=\frac{\text{y}+2}{-5}=\frac{\text{z}}{1}$ and $\frac{\text{x}}{1}=\frac{\text{y}}{2}=\frac{\text{z}}{3}$ are perpendicular to each other.
→
Integrate the following integrals:
$\int\cos3\text{x}\cos4\text{x dx}$
→