Find the angle between the following pairs of lines:
(i)\(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\)
(ii \(\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\) and \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\)
Find the angle between the following pairs of lines:
(i)\(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\)
(ii \(\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\) and \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\)
Solution and Explanation
(i) Let \(\overrightarrow b_1\) and \(\overrightarrow b_2\) be the vectors parallel to the pair of lines, \(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\), respectively.
∴ \(\overrightarrow b_1=2\hat i+5\hat j-3\hat k\) and \(\overrightarrow b_2=-\hat i+8\hat j+4\hat k\)
\(\mid \overrightarrow b_1\mid =\sqrt{(2)^2+(5)^2+(-3)^2}=\sqrt {38}\)
\(\mid \overrightarrow b_2\mid =\sqrt{(-1)^2+(8)^2+(4)^2}=\sqrt {81}=9\)
\(\mid \overrightarrow b_1.\mid \overrightarrow b_2\) = \((2\hat i+5\hat j-3\hat k)\).\((-\hat i+8\hat j+4\hat k)\)
= 2(-1)+5×8+(-3).4
=-2+40-12
=26
The angle, Q, between the given pair of lines is given by the relation,
cos Q=\(\begin{vmatrix}\frac{\overrightarrow b_1.\overrightarrow b_2}{\|\overrightarrow b_1 \| \overrightarrow b_2 \|} \end{vmatrix}\)
\(\Rightarrow\) cos Q=\(\frac{26}{9\sqrt{38}}\)
\(\Rightarrow\) Q=\(\cos^{-1} \bigg(\frac{26}{9\sqrt{38}}\bigg)\)
(ii) Let \(\overrightarrow b_1,\overrightarrow b_2\) be the vectors parallel to the given pair of lines, \(\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\) and \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\), respectively.
\(\overrightarrow b_1=2\hat i+2\hat j+\hat k\) and \(\overrightarrow b_2=4\hat i+\hat j+8\hat k\)
∴|\(\overrightarrow b_1\)|=\(\sqrt{(2)^2+(2)^2+(1)^2}=\sqrt9=3\)
|\(\overrightarrow b_2\)|=\(\sqrt{4^2+1^2+8^2}=\sqrt{81}=9\)
\(\overrightarrow b_1.\overrightarrow b_2\)=\((2\hat i+2\hat j+\hat k).\)\((4\hat i+\hat j+8\hat k)\)
=2×4+2×1+1×8
=8+2+8
=18
If Q is the angle between the given pair of lines,
then cos Q=\(\begin{vmatrix}\frac{\overrightarrow b_1.\overrightarrow b_2}{\|\overrightarrow b_1 \| \overrightarrow b_2 \|} \end{vmatrix}\)
\(\Rightarrow\) cos Q=\(\frac{18}{3*9}=\frac{2}{3}\)
\(\Rightarrow\) Q= \(\cos^{-1}\bigg(\frac{2}{3}\bigg)\)
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Three Dimensional Geometry Questions
Find the shortest distance between the lines \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}\) and \(\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}\)
Find the shortest distance between the lines whose vector equations are
\(\overrightarrow r=(\hat i+2\hat j+3\hat k)+\lambda(\hat i-3\hat j+2\hat k)\)
and \(\overrightarrow r=(4\hat i+5\hat j+6\hat k)+\mu(2\hat i+3\hat j+\hat k)\)
Find the shortest distance between the lines whose vector equations are
\(\overrightarrow r=(1-t)\hat i+(t-2)\hat j+(3-2t)\hat k\) and
\(\overrightarrow r=(s+1)\hat i+(2s-1)\hat j-(2s+1)\hat k\)
In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(a) (0,0,0) 3x-4y+12z=3
(b) (3,-2,1) 2x-y+2z+3=0
(c) (2,3,-5) x+2y-2z=9
(d) (-6,0,0) 2x-3y+6z-2=0determine whether the given planes are parallel or perpendicular,and in case they are neither, find the angles between them. (a)7x+5y+6z+30=0 and 3x-y-10z+4=0
(b)2x+y+3z-2=0 and x-2y+5=0
(c)2x-2y+4z+5=0 and 3x-3y+6z-1=0
(d)2x-y+3z-1=0 and 2x-y+3z+3=0
(e)4x+8y+z-8=0 and y+z-4=0
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
Concepts Used:
Angle between Two Lines
The two straight lines, whenever intersects, form two sets of angles. The angles so formed after the intersection are;
- A pair of acute angle
- Another pair of an obtuse angle
The absolute values of angles created depend on the slopes of the intersecting lines.
It is also worth taking note, that the angle so formed by the intersection of two lines cannot be calculated if any of the lines is parallel to the y-axis as the slope of a line parallel to the y-axis is an indeterminate.
Read More: Angle Between Two Lines