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Let the coordinates of the foot of perpendicular P from the origin to the plane be  Given plane equation . or written as  The direction ratios of the normal of the plane are . Therefore  So, now dividing both sides of the equation by  we will obtain, This equation is similar to  where,  are the directions cosines of normal to the plane and d is the distance of normal from the origin. Then...
Let the coordinates of the foot of perpendicular P from the origin to the plane be  Given plane equation . The direction ratios of the normal of the plane are . Therefore  So, now dividing both sides of the equation by  we will obtain, This equation is similar to  where,  are the directions cosines of normal to the plane and d is the distance of normal from the origin. Then finding the...
Let the coordinates of the foot of perpendicular P from the origin to the plane be  Given a plane equation , Or,  The direction ratios of the normal of the plane are . Therefore  So, now dividing both sides of the equation by  we will obtain, This equation is similar to  where,  are the directions cosines of normal to the plane and d is the distance of normal from the origin. Then finding the...
Let the coordinates of the foot of perpendicular P from the origin to the plane be  Given a plane equation , Or,  The direction ratios of the normal of the plane are . Therefore  So, now dividing both sides of the equation by  we will obtain, This equation is similar to  where,  are the directions cosines of normal to the plane and d is the distance of normal from the origin. Then finding the...
Given the equation of plane  So we have to find the Cartesian equation, Any point  on this plane will satisfy the equation and its position vector given by,   Hence we have,   Or,   Therefore this is the required Cartesian equation of the plane.
Given the equation of plane  So we have to find the Cartesian equation, Any point  on this plane will satisfy the equation and its position vector given by, Hence we have,   Or,   Therefore this is the required Cartesian equation of the plane.
Given the equation of the plane  So we have to find the Cartesian equation, Any point  on this plane will satisfy the equation and its position vector given by, Hence we have,   Or,   Therefore this is the required Cartesian equation of the plane.
We have given the distance between the plane and origin equal to 7 units and normal to the vector . So, it is known that the equation of the plane with position vector  is given by, the relation,  , where d is the distance of the plane from the origin. Calculating ;   is the vector equation of the required plane.
Given the equation of plane is   or we can write So, the direction ratios of normal from the above equation are, . Therefore  Then dividing both sides of the plane equation by , we get So, this is the form of   the plane, where  are the direction cosines of normal to the plane and d is the distance of the perpendicular drawn from the origin.  The direction cosines of the given line are  and...
Given the equation of plane is    So, the direction ratios of normal from the above equation are, . Therefore  Then dividing both sides of the plane equation by , we get So, this is the form of   the plane, where  are the direction cosines of normal to the plane and d is the distance of the perpendicular drawn from the origin.  The direction cosines of the given line are  and the distance of...
Given the equation of the plane is  or we can write  So, the direction ratios of normal from the above equation are, . Therefore  Then dividing both sides of the plane equation by , we get So, this is the form of   the plane, where  are the direction cosines of normal to the plane and d is the distance of the perpendicular drawn from the origin.  The direction cosines of the given line...
Equation of plane Z=2, i.e.   The direction ratio of normal is 0,0,1 Divide equation  by 1 from both side  We get,       Hence, direction cosins are 0,0,1. The distance of the plane from the origin is 2.
Given two equations of the line      in the vector form. So, we will apply the distance formula  for knowing the distance between two lines    and    After comparing the given equations, we obtain                                  Then calculating the determinant value numerator. That implies, Now, after substituting the value in the above formula we get, Therefore, units are the shortest...
Given two equations of line    in the vector form. So, we will apply the distance formula  for knowing the distance between two lines    and    After comparing the given equations, we obtain                                                   Then calculating the determinant value numerator. That implies,  Now, after substituting the value in the above formula we get, Therefore, is the...
We have given two lines:   and  Calculating the shortest distance between the two lines,  and  by the formula  Now, comparing the given equations, we obtain Then calculating determinant    Now calculating the denominator, So, we will substitute all the values in the formula above to obtain, Since distance is always non-negative, the distance between the given lines is   units.
So given equation of lines;   and   in the vector form. Now, we can find the shortest distance between the lines  and , is given by the formula, Now comparing the values from the equation, we obtain                                   Then calculating  So, substituting the values now in the formula above we get; Therefore, the shortest distance between the two lines is  units.
First, we have to write the given equation of lines in the standard form;   and  Then we have the direction ratios of the above lines as;    and      respectively.. Two lines with direction ratios  and   are perpendicular to each other if,   Therefore the two lines are perpendicular to each other.
First we have to write the given equation of lines in the standard form;    and   Then we have the direction ratios of the above lines as;    and      respectively.. Two lines with direction ratios  and   are perpendicular to each other if,   Thus, the value of p is .
Given lines are;     and  So, we two vectors  which are parallel to the pair of above lines respectively.    and   To find the angle A between the pair of lines  we have the formula; Then we have and  Therefore we have; or
Given lines are;   and   So, we two vectors  which are parallel to the pair of above lines respectively.    and   To find the angle A between the pair of lines  we have the formula; Then we have and  Therefore we have; or
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