Question

Find the Cartesian equations of the line which passes through the point (2, 1, 3) and perpendicular to the lines `(x - 1)/(1) = (y - 2)/(2) = (z - 3)/(3) and x/(-3) = y/(2) = z/(5)`.

Answer 1

Let line L₁ be   `(x - 1)/(1) = (y - 2)/(2) = (z - 3)/(3)`

Let line L₂ be  `x/(-3) = y/(2) = z/(5)`

It is perpendicular to the vector `bar"b" = hat"i" + 2hat"j" + hat"k" and bar"c" = -3hat"i" + 2hat"j" + 5hat"k"`.

∴ It is perpendicular to lines whose direction ratios are 1, 2, 1 and -3, 2, 5.

Now, `bar"b"xxbar"c"=|(hat"i",hat"j",hat"k"),(1,2,3),(-3,2,5)|`

`=hat"i"(10-6)-hat"j"(5+9)+hat"k"(2+6)`

`bar"b"xxbar"c"= 4hat"i"-14hat"j"+8hat"k"`

∴ Is parallel to the required line.

The direction ratios of the parallel line are 4, -14, 8 or 2, -7, 4 

Let the required line passing through point A = (2, 1, 3) = (x₁ y₁ z₁)

∴  The cartesian equations of the line is 

`(x - x_1)/a = (y - y_1)/b = (z - z_1)/c`

 `(x - 2)/(2) = (y - 1)/(-7) = (z - 2)/(4)`