Mathematics Angle Between two intersection lines, Planes
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Angle Between two intersection lines, Planes

Angle between two lines :

If direction ratios of two lines are `a_1, b_1, c _1` and `a_2, b_2, c_2` then acute angle between two lines is given by

`quad quad quad quad quad quad cos theta = |a_1a_2 +b_1b_2 +c_1c_2 |/ (sqrt (a_1^2+b_1^2 +c_1^2) sqrt (a_2^2+b_2^2+c_2^2))`

`text( Proof:)` Vector along lines can be taken as `hat (a) = a_1 hat (i)+ b_1hat(j) + c_1 hat(k)` and `hat (b) = a _2hat (i) + b_2hat (j) + c_2hat (k)`.

Acute angle between lines` = `acute angle between vectors `veca` and `vecb`.

`quad quad quad quad :.` `cos theta = |vec(a)*vec(b) |/( |vec(a) | |vec(b) |) = (a_1a_2+b_1b_2+c_1c_2)/ (sqrt (a_1^2+b_1^2 +c_1^2) sqrt (a_2^2+b_2^2+c_2^2))`


If direction cosines of lines are `l_1,m_1,n_1`and `l_2, m_2, n_2` then acute angle between them is given by

`quad quad quad quad quad cos theta= | l_1l_2 +m_1m_2 +n_1n_2 |`.

`text(Note :)`

`(i)` If lines are perpendiculars (i.e. vectors along them are also perpendicular) the

`quad quad quad quad quad a _1a_2 + b_ 1b_2 + c_ 1c_2 = 0` or `l_1l_2 + m_1m_2 + n_1n_2 = 0`.

`(ii)` lf lines are parallel (i.e. vectors along them area also parallel) then `a_1/a_2= b_1/b_2=c_1/c_2 ` or `l_1/l_2 =m_1/m_2 =n_1/n_2`

Angle Between Two Planes :

`text(1. Vector form :)`

The angle between the two planes is defined as the angle between their normals.

Let `theta` be the angle between planes;

`vec(r)*vec(n_1) =d_1 ` and `vec(r)*vec(n_2) =d_2` is given by

`quad quad quad quad quad cos theta = (vec(n_1)*vec(n_2)) / ( | vec(n_1)| | vec(n_2) | )`

`text(2. Cartesian form :)`

The angle `theta` between the planes `a_1x +b_1 y +c_1z +d_1 =0` and `a_2x +b_2y+c_2z+d_2=0` is given by


`quad quad quad quad cos theta = (a_1a_2+b_1b_2+c_1c_2)/sqrt((a_1^2+b_1^2+c_1^2)sqrt(a_2^2+b_2^2+c_2^2))`

`text(3.Two planes are perpendicular)` iff

`quad quad quad quad a_1a_2 + b_1b_2 + c_1c_2 = 0` & Parallel if `vec(n_1) xx vec(n_2) =0`
 
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