Q. 21

A line passes through (2, –1, 3) and is perpendicular to the lines and Obtain its equation in vector and cartesian form.

Answer :


To write the equation of a line, we need a point on the line which is given (2, –1, 3) and direction of the line


We need to find the equation of the red line as shown


So we need to find the direction of the red line


The red line is perpendicular to (Blue line) and (orange line)


So we have to take the cross product of directions of blue and orange line in order to get the direction of the red line


Compare the given vector equations of orange and blue lines with standard form where is the direction of line and is point on line


Here the given directions are (blue) and (orange)


Take the cross product, and we will get the direction of the red line which is perpendicular to these






Now for the red line


We have a point on it (2, –1, 3) and also its direction


Hence the equation of a line in vector form will be



The general cartesian form of the equation is


Where (a1, a2, a3) is point on line and <b1, b2, b3> is the direction ratio


And hence the cartesian form of the equation is




Hence the vector and cartesian form of the required line is


and respectively.


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