Angle Between Line and Plane
Category : JEE Main & Advanced
The angle \[\theta \] between the line \[\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}\], and the plane \[ax+by+cz+d=0\], is given by \[\sin \theta =\frac{al+bm+cn}{\sqrt{({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}\sqrt{({{l}^{2}}+{{m}^{2}}+{{n}^{2}})}}\].
(i) The line is perpendicular to the plane if and only if \[\frac{a}{l}=\frac{b}{m}=\frac{c}{n}\].
(ii) The line is parallel to the plane if and only if \[al+bm+cn=0\].
(iii) The line lies in the plane if and only if \[al+bm+cn=0\] and \[a\alpha +b\beta +c\gamma +d=0\].
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