Download App

Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry4 Marks
Question
Find the angle between the lines whose direction cosines are given by the equations:
2l - m + 2n = 0 and mn + nl + lm = 0
✓

Answer

Given that,
2l - m + 2n = 0 .....(1)
mn + nl + lm = 0 .....(2)
From equation (1),
2l - m + 2n = 0
m = 2l + 2n  
Put the value of m in equation (2),
mn + nl + lm = 0
(2l + 2n) n + nl + l(2l + 2n) = 0
2ln + 2n2 +nl +2l2 + 2ln = 0
2l2 + 5ln + 2n2 = 0
2l2 + 4ln + ln + 2ln2 = 0
2l (l + 2n) + n(l + 2n) = 0
(1 + 2n) (2l = n) = 0
l + 2n = 0 or 2l + n = 0
l = -2n or  $\text{l}=-\frac{\text{n}}{2}$
Put the value of l = -2n in equation (1)
2l - m + 2n = 0
2 (-2n) - m + 2n = 0
-4n - m + 2n = 0
-2n - m = 0
-2n = m
m = -2n
Again, put the value of $\text{l}=-\frac{1}{2}$ in equation (1)
2l - m + 2n = 0
$2\Big(-\frac{1}{2}\text{n}\Big)-\text{m}+2\text{n}=0$
-n - m + 2n = 0
-m + n = 0
-m = -n
m = n
So, direction cosines of the lines are given by,
-2n, -2n, n or $-\frac{1}{2},\text{n},\text{n},\text{n}$
-2, -2, 1 or $-\frac{1}{2},1,1$
So, vectors parallel to these lines
$\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=-\frac{1}{2}\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ respectively.
Let, $\theta$ be the angle between the $\vec{\text{a}}$ and $\vec{\text{b}},$
 $\cos\theta=\frac{\vec{\text{a}}\times\vec{\text{b}}}{\big|\vec{\text{a}}\big|\big|\vec{\text{b}}\big|}$
$=\frac{\big(-2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}\big)\times\Big(-\frac{1}{2}\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\Big)}{\sqrt{(-2)^2+(-2)^2+(1)^2}\sqrt{\Big(-\frac{1}{2}^2\Big)+(1)^2+(1)^2}}$
$=\frac{(-2)\big(-\frac{1}{2}\big)+(-2)(1)+(1)(1)}{\sqrt{4+4+1}\sqrt{\frac{1}{4}1+1}}$
$=\frac{1-2+1}{\sqrt{9}\sqrt{\frac{9}{4}}}$
$=\frac{0}{3\times\frac{3}{2}}$
$\cos\theta=0$
$\theta=\cos^{-1}(0)$
 $\theta=\frac{\pi}{2}$
So, angle between the lines $=\frac{\pi}{2}$.

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 Three Dimensional Geometry→Three Dimensional Geometry — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the area, lying above x-axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x.
→
Evaluate:

$\int\limits_0^\frac{\pi}{4} \sin 2x \sin 3\text{x dx}$

→
Find the equation of the plane that contains the point (1, -1, 2) and is perpendicular to each of the planes 2x + 3y - 2z = 5 and x + 2y - 3z = 8.
→
Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=(\text{x}-1)\text{e}^{\text{x}}$
→
Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
→
Using properties of determinants, show that

$\Delta=\begin{vmatrix} \text{b+c} & \text{a} & \text{a} \\ \text{b} & \text{c+a} & \text{b} \\ \text{c} & \text{c} & \text{a+b} \end{vmatrix}=\text{4 abc}.$

→
If the derivative of tan-1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b.
→
An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made, on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However at least 4 times as many passengers prefer to travel by economy class than by executive class. Formulate linear programming problem in order to maximize the profit for the airline.
→
Find the area between the curves y = x and y = x2.
→
If $x=a \cos \theta+b \sin \theta$ and $y=a \sin \theta-b \cos \theta$ then prove that $y^2 y_2-x y_1+y=0$.
→