Find the value of λ, so that the lines and are at right angles. Also, find whether the lines are intersecting or not.

Question

Find the value of λ, so that the lines  and  are at right angles. Also, find whether the lines are intersecting or not.

Philoid · Accepted Answer

The direction ratios of a line, if the cartesian equation of the line is given , is a,b,c.

Hence, the direction ratios of line L:

and the direction ratios of line M:

two lines with direction ratios, a₁, a₂, a₃ and b₁, b₂, b₃, are perpendicular if and only if a₁b₁+a₂b₂+a₃b₃=0

Hence,

Solving this equation, we get

Hence, the equation of lines are:

Any general point on line will be

and on the line will be .

For the lines to intersect, . Solving the equations for x and y-coordinates,

we get and m=1.

Equating equations for z-coordinates, we get

3k+3 = 5m+6

Substituting and m=1, we get

or 15=22 which is not true.

Hence, the lines do not intersect in real space.