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

Question

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

Philoid · Accepted 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.