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 \(\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}\) and \(\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}\)
Find the shortest distance between the lines\(\overrightarrow r=(\hat i+2\hat j+\hat k)+\lambda(\hat i-\hat j+\hat k)\) and\(\overrightarrow r=2\hat i-\hat j-\hat k+\mu\,(2\hat i+2\hat j+2\hat k)\)
Find the vector equation of the line passing through the point (1, 2, -4)and perpendicular to the two lines: \(\frac {x-8}{3}=\frac {y+19}{-16}=\frac {z-10}{7}\) and \(\frac {x-15}{3}=\frac {y-29}{8} =\frac {z-5}{-5}\)
If l1,m1,n1 and l2,m2,n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2-m2n1,n1l2-n2l1,l1m2-l2m1.
Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then \(\frac {1}{a^2}+\frac {1}{b^2}+\frac {1}{c^2} =\frac {1}{p^2}\).
Distance between the two planes: 2x+3y+4z = 4 and 4x+6y+8z = 12 is
\(The\ planes: 2x-y+4z=5 \ and \ 5x-2.5y+10z=6\ are\)
Find the angle between the planes whose vector equations are
\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=5\) and \(\overrightarrow r.(3\hat i-3\hat j+5\hat k)=3\)
Find the equation of the plane through the line of intersection of the planesx+y+z=1 and 2x+3y+4z=5 which is perpendicular to the plane x-y+z= 0
The cartesian equation of a line is \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\). Write its vector form.
Find the cartesian equation of the line that passes through the point(-2,4,-5)and parallel to the line given by \(\frac{x+3}{3}\)=\(\frac{y-4}{5}\)=\(\frac{z+8}{6}\)
Find the vector equation of the plane passing through the intersection of the planes
\(\overrightarrow r.(2\hat i+2\hat j-3\hat k)=7\),\(\overrightarrow r.(2\hat i+5\hat j+3\hat k)=9\) and through the point ( 2, 1, 3 ).
Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2\(\hat i\)-\(\hat j\)+4\(\hat k\) and is in the direction \(\hat i\)+2\(\hat j\)-\(\hat k\).
Find the equation of the plane through the intersection of the planes3x-y+2z-4 =0 and x+y+z-2=0 and the point (2, 2, 1).
Find the equation of the line that passes through the point(1,2,3)and is parallel to the vector 3\(\hat i\)+2\(\hat j\)-2\(\hat k\).
Find the intercepts cut off by the plane 2x+y-z = 5
Find the equation of the planes that passes through three points.
(a) (1,1,-1),(6,4,-5),(-4,-2,-3)
(b) (1,1,0),(1,2,1),(-2,2,-1).
Show that the three lines with direction cosines\(\frac{12}{13}\),\(-\frac{3}{13}\),\(-\frac{4}{13}\) ; \(\frac{4}{13}\),\(\frac{12}{13}\),\(\frac{3}{13}\);\(\frac{3}{13}\),\(-\frac{4}{13}\),\(\frac{12}{13}\) are mutually perpendicular.
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 \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\)
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) 2x+3y+4z-12=0 (b) 3y+4z-6=0
(c)x+y+z=1 (d) 5y+8=0