Sir/Madam In the problem ,Find the value of lambda Such that the line (X-2)/9 = Y-1/(lambda) = (Z+3)/-6 is perpendicular - Maths - Three Dimensional Geometry

Question

Sir/Madam
In the problem ,Find the value of lambda Such that the line
(X-2)/9 = Y-1/(lambda) = (Z+3)/-6 is perpendicular to the plane
3X-Y-2Z=7 (ans Lambda = - 3)(since line is perpendicular to the
plane & parallel to its normal) How do we find the equation of
the normal to the given Plane?

Ankush Jain · Accepted Answer

From Cartesian form ,we have, if Î¸ is angle between the line $\frac{x - x_{1}}{l} = \frac{y - y_{1}}{m} = \frac{z - z_{1}}{n}$ and the plane ax + by + cz + d = 0, then $\sin θ = \frac{a l + b m + c n}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{l^{2} + m^{2} + n^{2}}}$ Now, condition for perpendicularity states that,if a given line is perpendicular to the plane, then it is parallel to its normal Therefore, $\vec{b} = l \vec{i} + m \vec{j} + n \vec{k} a n d \vec{n} = a \vec{i} + b \vec{j} + c \vec{k} a r e p a r a l l e l$ Consequently, we have $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$ Now, the given equation of the line is: $\frac{x - 2}{9} = \frac{y - 1}{λ} = \frac{z + 3}{- 6}$ and equation of the plane is: 3x - y - 2z = 7 As, it is given that line is parallel to plane, therefore, we have $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$ On putting the respective values of l, m, n, a , b and c in above equation, we get $\frac{9}{3} = \frac{λ}{- 1} = \frac{- 6}{- 2} \Rightarrow 3 = - λ = 3 \Rightarrow - λ = 3 \Rightarrow λ = - 3$ Hope you get it!!

Sir/Madam In the problem ,Find the value of lambda Such that the line (X-2)/9 = Y-1/(lambda) = (Z+3)/-6 is perpendicular to the plane 3X-Y-2Z=7. (ans Lambda = - 3)(since line is perpendicular to the plane & parallel to its normal).How do we find the equation of the normal to the given Plane?

Sir/Madam

In the problem ,Find the value of lambda Such that the line (X-2)/9 = Y-1/(lambda) = (Z+3)/-6 is perpendicular to the plane

3X-Y-2Z=7. (ans Lambda = - 3)(since line is perpendicular to the plane & parallel to its normal).How do we find the equation of the normal to the given Plane?