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D Divya Prakash Singh
Given direction ratios   and  . Thus the angle between the lines A is given by; a Thus, the angle between the lines is .

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D Divya Prakash Singh
Given that  are the direction cosines of two mutually perpendicular lines. Therefore, we have the relation:                               .........................(1)          .............(2) Now, let us assume  be the new direction cosines of the lines which are perpendicular to the line with direction cosines. Therefore we have,  Or,      ......(3) So, l,m,n are the direction cosines of the...

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D Divya Prakash Singh
We can assume the line joining the origin, be OA where  and the point  and PQ be the line joining the points  and . Then the direction ratios of the line OA will be   and that of line PQ will be So to check whether line OA is perpendicular to line PQ then, Applying the relation we know, Therefore OA is perpendicular to line PQ.

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D Divya Prakash Singh
We know that the distance between a point  and a plane  is given by,                                    .......................(1) So, calculating for each case; (a) Point  and Plane  Therefore,  (b) Point  and Plane  Therefore,  (c) Point  and Plane  Therefore,  (d) Point  and Plane  Therefore,   

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D Divya Prakash Singh
Two planes  whose direction ratios are  and  whose direction ratios are , are said to Parallel:  If,  and Perpendicular: If,  And the angle between  is given by the relation, So, given two planes  Here,    and    So, applying each condition to check: Parallel check:    Clearly, the given planes are NOT parallel as  . Perpendicular check:  . Clearly, the given planes are...

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D Divya Prakash Singh
Two planes  whose direction ratios are  and  whose direction ratios are , are said to Parallel:  If,  and Perpendicular: If,  And the angle between  is given by the relation, So, given two planes  Here,    and    So, applying each condition to check: Parallel check:    Therefore  Thus, the given planes are parallel to each other.

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D Divya Prakash Singh
Two planes  whose direction ratios are  and  whose direction ratios are , are said to Parallel:  If,  and Perpendicular: If,  And the angle between  is given by the relation, So, given two planes  Here,    and    So, applying each condition to check: Parallel check:    Thus, the given planes are parallel as  .

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D Divya Prakash Singh
Two planes  whose direction ratios are  and  whose direction ratios are , are said to Parallel:  If,  and Perpendicular: If,  And the angle between  is given by the relation, So, given two planes  Here,    and    So, applying each condition to check: Perpendicular check:  . Thus, the given planes are perpendicular to each other.  

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D Divya Prakash Singh
Two planes  whose direction ratios are  and  whose direction ratios are , are said to Parallel:  If,  and Perpendicular: If,  And the angle between  is given by the relation, So, given two planes  Here,    and    So, applying each condition to check: Parallel check:    Clearly, the given planes are NOT parallel. Perpendicular check:  . Clearly, the given planes are NOT perpendicular. Then...

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D Divya Prakash Singh
Given two vector equations of plane  and . Here,   and   The formula for finding the angle between two planes,                                          .............................(1) and        Now, we can substitute the values in the angle formula (1) to get, or   or  

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D Divya Prakash Singh
The equation of the plane through the intersection of the given two planes,   and   is given in Cartesian form as; or              ..................(1) So, the direction ratios of (1) plane are  which are . Then, the plane in equation (1) is perpendicular to  whose direction ratios  are . As planes are perpendicular then, we get, or   or   Then we will substitute the values of  in the...

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D Divya Prakash Singh
Here  and   and   and   Hence, using the relation , we get or                  ..............(1) where,  is some real number. Taking , we get or    or                          .............(2) Given that the plane passes through the point , it must satisfy (2), i.e.,                   or                 Putting the values of  in (1), we get or        or       which is the required vector...

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D Divya Prakash Singh
The equation of any plane through the intersection of the planes, Can be written in the form of; , where  So, the plane passes through the point , will satisfy the above equation. That implies  Now, substituting the value of  in the equation above we get the final equation of the plane;  is the required equation of the plane.    

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D Divya Prakash Singh
Given that the plane is parallel to the ZOX plane. So, we have the equation of plane ZOX as  . And an intercept of 3 on the y-axis  Intercept form of a plane given by; So, here the plane would be parallel to the x and z-axes both. we have any plane parallel to it is of the form, . Equation of the plane required is .  

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D Divya Prakash Singh
Given plane   We have to find the intercepts that this plane would make so, Making it look like intercept form first: By dividing both sides of the equation by 5 (as we have to make the R.H.S =1) , we get then, So, as we know that from the equation of a plane in intercept form, where a,b,c are the intercepts cut off by the plane at x,y, and z-axes respectively. Therefore after comparison, we...

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D Divya Prakash Singh
The equation of the plane which passes through the three points   is given by; Determinant method, As determinant value is not equal to zero hence there must be a plane that passes through the points A, B, and C. Finding the equation of the plane through the points,  After substituting the values in the determinant we get, So, this is the required Cartesian equation of the plane.

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D Divya Prakash Singh
The equation of the plane which passes through the three points   is given by; Determinant method, Or,  Here, these three points A, B, C are collinear points. Hence there will be an infinite number of planes possible which passing through the given points.  

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S seema garhwal
   (a)      (1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3)                                                                      A,B,C are collinear points so there are infinite number of the planes that passes through three points

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D Divya Prakash Singh
Given the point  and the normal vector  which is perpendicular to the plane is  The position vector of point A is  So, the vector equation of the plane would be given by, Or  where  is the position vector of any arbitrary point  in the plane.   Therefore, the equation we get,      So, this is the required Cartesian equation of the plane.    

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D Divya Prakash Singh
Given the point  and the normal vector  which is perpendicular to the plane is  The position vector of point A is  So, the vector equation of the plane would be given by, Or  where  is the position vector of any arbitrary point  in the plane.   Therefore, the equation we get,     or    So, this is the required Cartesian equation of the plane.
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