## Three - dimensional Geometry

# Angle between Two Lines

- The angle between the two lines whose direction ratios are given by (a
_{1}, b_{1}, c_{1}) and (a_{2}, b_{2}, c_{2}) is \cos\theta=\frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}

Lines are parallel \Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

Lines are perpendicular ⇒ a_{1}a_{2 }+ b_{1}b_{2 }+ c_{1}c_{2 }= 0 - The angle between the two lines whose direction cosines are given by (l
_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2}) is cos θ = |l_{1}l_{2 }+ m_{1}m_{2 }+ n_{1}n_{2}|

Lines are parallel \Rightarrow \frac{l_{1}}{l_{2}}=\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}

Lines are perpendicular ⇒ l_{1}l_{2 }+ m_{1}m_{2 }+ n_{1}n_{2 }= 0 - If (l
_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2}) are the direction cosines of two concurrent lines, then direction cosines of lines bisecting the angles between them are (l_{1 }± l_{2, }m_{1 }± m_{2,}n_{1 }± n_{2}). - The direction cosines of angular bisectors between the lines whose direction cosines are given by (l
_{1}, m_{1}, n_{1}) and (l_{2}, m_{2}, n_{2}) and θ is an angle between them is

\frac{l_{1}+l_{2}}{2 \cos \frac{\theta}{2}},\frac{m_{1}+m_{2}}{2 \cos \frac{\theta}{2}},\frac{n_{1}+n_{2}}{2 \cos \frac{\theta}{2}} (or) \frac{l_{1}-l_{2}}{2 \sin \frac{\theta}{2}},\frac{m_{1}-m_{2}}{2 \sin \frac{\theta}{2}},\frac{n_{1}-n_{2}}{2 \sin \frac{\theta}{2}} - If α, β, γ, δ are the angles made by a line with the four diagonals of a cube, then

cos^{2}α + cos^{2}β + cos^{2}γ + cos^{2}δ = \frac{4}{3}

sin^{2}α + sin^{2}β + sin^{2}γ + sin^{2}δ = \frac{8}{3} - The angle between any two diagonals of the cube is \cos^{-1}\left(\frac{1}{3}\right).
- The angle between diagonal of a cube and diagonal of a face of a cube is \cos^{-1}\left(\sqrt{\frac{2}{3}}\right)
- The angle between diagonal of cube and edge of a cube is \cos^{-1}\left(\frac{1}{\sqrt{3}}\right).
- If the edges of a rectangular parallelopiped are a, b, c then the angle between four diagonals are \cos^{-1}\left(\frac{|1\pm a^{2}\pm b^{2}\pm c^{2}|}{a^{2}+b^{2}+c^{2}}\right)

(In Numerator all three signs are not positive (or) not negative) - The 'θ' is angle between the lines \vec{r}=\vec{a}_{1}+\lambda \vec{b}_{1} , \vec{r}=\vec{a}_{2}+\mu \vec{b}_{2} then \cos \theta=\frac{\vec{b}_{1}\cdot \vec{b}_{2}}{|\vec{b}_{1}|\ |\vec{b}_{2}|}
- If θ is angle between the lines \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}} and \frac{x-x_{2}}{a_{2}}=\frac{y-y_{2}}{b_{2}}=\frac{z-z_{2}}{c_{2}} then \cos \theta=\frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_1^2+b_1^2+c_1^2}\ \sqrt{a_2^2+b_2^2+c_2^2}}

### View the Topic in this video From 31:18 To 37:18

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1. If l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2 }are the direction cosines of two lines; and θ is the acute angle between the two lines; then cos θ = |l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}|

2. If a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2 }are the direction ratios of two lines and θ is the acute angle between the two lines; then \cos \theta=\begin{vmatrix}\frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}} \end{vmatrix}

3. If θ is the acute angle between \vec{r}=\vec{a}_{1}+\lambda \vec{b}_{1} and \vec{r}=\vec{a}_{2}+\lambda \vec{b}_{2}, then \cos \theta=\begin{vmatrix}\frac{\vec{b}_{1} \cdot \vec{b}_{2}}{|\vec{b}_{1}| \ |\vec{b}_{2}|} \end{vmatrix}

4. If \frac{x-x_{1}}{l_{1}}=\frac{y-y_{1}}{m_{1}}=\frac{z-z_{1}}{n_{1}} and \frac{x-x_{2}}{l_{2}}=\frac{y-y_{2}}{m_{2}}=\frac{z-z_{2}}{n_{2}} are the equations of two lines, then the acute angle between the two lines is given by cos θ = |l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}|.