Download App

Three Dimensional Geometry — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSThree Dimensional Geometry5 Marks
Question
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
$4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$
✓

Answer

The direction ratios of normal to the plane, $L_1: a_1x + b_1y + c_1z = 0,$
are $a_1, b_1, c_1$ and $L_2: a_2x + b_2y + c_2z = 0$ are $a_2, b_2, c_2$
$\text{L}_1||\text{L}_2,\ \text{if }\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}=\frac{\text{c}_1}{\text{c}_2}$
$\text{L}_1\perp\text{L}_2,\ \text{if}\ \text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2=0$
The angle between $L_1$ and $L_2$ is given by,
$\text{Q}=\cos^{-1}\Bigg|\frac{\text{a}_1\text{a}_2+\text{b}_1\text{b}_2+\text{c}_1\text{c}_2}{\sqrt{\text{a}_1^2+\text{b}_1^2+\text{c}_1^2}.\sqrt{\text{a}_2^2+\text{b}_2^2+\text{c}_2^2}}$
The equations of the given planes are $4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$
Here, $a_1 = 4, b_1 = 8, c_1 = 1$ and $a_2 = 0, b_2 = 1, c_2 = 1$
$a_1a_2 +b_1b_2 + c_1c_2 = 4\times 0 + 8 \times 1 + 1$
$=9\neq0$
$\frac{\text{a}_1}{\text{a}_2}=\frac{4}{0},\ \frac{\text{b}_1}{\text{b}_2}=\frac{8}{1}=8\text{ and } \frac{\text{c}_1}{\text{c}_2}=\frac{1}{1}=1$
$\therefore\ \ \frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
Therefore, the given lines are not parallel to each other.
The angle between the planes is given by,
$\text{Q}=\cos^{-1}\Bigg|\frac{4\times0+8\times1+1\times1}{\sqrt{4^2+8^2+1^2}\times\sqrt{0^2+1^2+1^2}}\Bigg|=\cos^{-1}\Bigg|\frac{9}{9\sqrt{2}}\Bigg|$
$=\cos^{-1}\Big(\frac{1}{\sqrt{2}}\Big)=45^{\circ}.$

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 "5 Marks Questions" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Solve the following systems of linear equations by cramer's rule:
x + 2y = 1,
3x + y = 4
→
Find which of the function in Exercises 2 to 10 is continuous or discontinuos at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\text{e}^{\frac{1}{\text{x}}}}{1+\text{e}^{\frac{1}{\text{x}}}},&\text{if x}\neq0\\0,&\text{if x }=\text{a}\end{cases}$
at x = 0
→
A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost $24.$ It takes $1$ hour to make a ring and $30$ minutes to make a chain. The maximum number of hours available per day is $16.$ If the profit on a ring is $Rs. 300$ and that on a chain is $Rs. 190,$ find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an $L.P.P.$ and solve it graphically.
→
Prove that the diagonals of a rhombus are perpendicular bisectors of each other.
→
Find the vector and Cartesian equations of the line through the point (1, 2, – 4) and perpendicular to the two lines.
$\overrightarrow{\text{r}} = (8\hat{\text{i}} - 19\hat{\text{j}} + 10\hat{\text{k}})+\lambda(3\hat{\text{i}} - 16\hat{\text{j}} + 7\hat{\text{k})}$ and $\overrightarrow{\text{r}} = (15\hat{\text{i}} - 29\hat{\text{j}} + 5\hat{\text{k}})+\mu(3\hat{\text{i}} - 8\hat{\text{j}} + 5\hat{\text{k})}.$
→
Integrate the function: $\frac{1}{9 x^{2}+6 x+5}$
→
Find the area of the minor segment of the circle $x^2+ y^2 = a^2$ cut off by the line $\text{x}=\frac{\text{a}}{2}.$ 
→
A company produces two types of leather belts, say type $A$ and $B.$ Belt $A$ is a superior quality and belt $B$ is of a lower quality. Profits on each type of belt are $Rs. 2$ and $Rs. 1.50$ per belt, respectively. Each belt of type $A$ requires twice as much time as required by a belt of type $B.$ If all belts were of type $B,$ the company could produce $1000$ belts per day. But the supply of leather is sufficient only for $800$ belts per day $($both $A$ and $B$ combined$).$ Belt $A$ requires a fancy buckle and only $400$ fancy buckles are available for this per day. For belt of type $B,$ only $700$ buckles are available per day.
How should the company manufacture the two types of belts in order to have a maximum overall profit?
→
Of the students in a college, it is known that $60\%$ reside in hostel and $40\%$ are day scholars $($not residing in hostel$).$ Previous year results report that $30\%$ of all students who reside in hostel attain $'A\ '$ grade and $20\%$ of day scholars attain $'A\ '$ grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an $'A\ '$ grade, what is the probability that the student is a hostlier?
→
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.
→