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D Divya Prakash Singh
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...

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D Divya Prakash Singh
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...

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D Divya Prakash Singh
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.  

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D Divya Prakash Singh
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.      

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D Divya Prakash Singh
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.

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D Divya Prakash Singh
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 .  

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D Divya Prakash Singh
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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D Divya Prakash Singh
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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D Divya Prakash Singh
To find the angle A between the pair of lines  we have the formula; We have two lines :  and The given lines are parallel to the vectors ; where    and    respectively, Then we have and  Therefore we have; or   

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D Divya Prakash Singh
To find the angle A between the pair of lines  we have the formula; We have two lines :  and The given lines are parallel to the vectors ; where    and    respectively, Then we have and  Therefore we have; or 

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D Divya Prakash Singh
Let the line passing through the points  and  is AB; Then as AB passes through through A so, we can write its position vector as; Then direction ratios of PQ are given by, Therefore the equation of the vector in the direction of AB is given by, We have then the equation of line AB in  vector form is given by, So, the equation of AB in Cartesian form is; or   

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D Divya Prakash Singh
GIven that the line is passing through the  and  Thus the required line passes through the origin.   its position vector is given by, So, the direction ratios of the line through   and  are, The line is parallel to the vector given by the equation,  Therefore the equation of the line passing through the point with position vector  and parallel to  is given by; Now, the equation of the line...

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D Divya Prakash Singh
Given the Cartesian equation of the line; Here the given line is passing through the point .  So, we can write the position vector of this point as; And the direction ratios of the line are 3, 7, and 2. This implies that the given line is in the direction of the vector, . Now, we can easily find the required equation of line: As we know that the line passing through the position vector  and...

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D Divya Prakash Singh
Given a line which passes through the point (– 2, 4, – 5) and is parallel to the line given by the ; The direction ratios of the line,  are 3,5 and 6. So, the required line is parallel to the above line. Therefore we can take direction ratios of the required line as 3k, 5k, and 6k, where k is a non-zero constant. And we know that the equation of line passing through the point  and with...

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D Divya Prakash Singh
Given that the line is passing through the point with position vector  and is in the direction of the line . And we know the equation of the line which passes through the point with the position vector and parallel to the vector  is given by the equation, So, this is the required equation of the line in the vector form. Eliminating , from the above equation we obtain the equation in the...

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D Divya Prakash Singh
We have given points where the line is passing through it; Consider the line joining the points (4, 7, 8) and (2, 3, 4) is AB and line joining the points  (– 1, – 2, 1) and (1, 2, 5)..is CD. So, we will find the direction ratios of the lines AB and CD; Direction ratios of AB are     or   Direction ratios of CD are     or  . Now, lines AB and CD will be parallel to each other if  Therefore we...

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D Divya Prakash Singh
It is given that the line is passing through A (1, 2, 3) and is parallel to the vector  We can easily find the equation of the line which passes through the point A and is parallel to the vector  by the known relation; , where  is a constant. So, we have now,   Thus the required equation of the line.

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D Divya Prakash Singh
We have given points where the line is passing through it; Consider the line joining the points (1, – 1, 2) and (3, 4, – 2) is AB and line joining the points  (0, 3, 2) and (3, 5, 6).is CD. So, we will find the direction ratios of the lines AB and CD; Direction ratios of AB are     or   Direction ratios of CD are     or  . Now, lines AB and CD will be perpendicular to each other...

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D Divya Prakash Singh
GIven direction cosines of the three lines;                And we know that two lines with direction cosines    and   are perpendicular to each other, if  Hence we will check each pair of lines: Lines ;  the lines  are perpendicular. Lines ;  the lines  are perpendicular. Lines ;  the lines  are perpendicular. Thus, we have all lines are mutually perpendicular to each other.
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