parametric equation

Question

Find the parametric equation of the line passing thru the point A(1,0,-1) that is perpendicular to the lines r₁= <3+2t, 3t, -4t> and r₂= <t, t, -t>.

 

Thank you

Answer 1

r₁ = <3 + 2t, 3t, -4t> = <3, 0, 0> + t<2, 3, -4>, v₁ = 2i + 3j - 4kr₂ = <t, t, -t> = <0, 0, 0> + t<1, 1, -1>, v₂ = i  + j - k

 

Let v be a vector perpendicular to both vectors v₁ and v_2.

This can be done by v = v₁ x v₂

 

v = |i  j  k|  This is a determinant.

      |2 3 -4|

      |1 1 -1|

    = (-3+ 4)i (-4+2)j + (2-3)k

    = i -2j - k

 

Then the vector perpendicular to r1 and r2 going through the

point (1, 0, -1) is r3 = <1, 0, -1> + t<1, -2, -1>

                              = <1+t, -2t, -1-t>