Find the angle between the lines whose direction ratios are a,b,c and b-c,c-a,a-b.
Solution and Explanation
The angle Q between the lines with direction cosines, a,b,c and b-c, c-a, a-b, is given by,
cosQ=|\(\frac{a(b-c)+b(c-a)+c(a-b)}{\sqrt{a^2+b^2+c^2}+\sqrt{(b-c)^2+(c-a)^2+(a-b)^2}}\)|
⇒cosQ=0
⇒Q=cos-1=0
⇒Q=90°
Thus, the angle between the lines is 90°.
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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}.
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- 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}
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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
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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
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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
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(e)4x+8y+z-8=0 and y+z-4=0
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