# Angle between Two Lines

• The angle between the two lines whose direction ratios are given by (a1, b1, c1) and (a2, b2, c2) 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 ⇒ a1a2 + b1b2 + c1c2 = 0
• The angle between the two lines whose direction cosines are given by (l1, m1, n1) and (l2, m2, n2) is cos θ = |l1l2 + m1m2 + n1n2|
Lines are parallel \Rightarrow \frac{l_{1}}{l_{2}}=\frac{m_{1}}{m_{2}}=\frac{n_{1}}{n_{2}}
Lines are perpendicular ⇒ l1l2 + m1m2 + n1n2 = 0
• If (l1, m1, n1) and (l2, m2, n2) are the direction cosines of two concurrent lines, then direction cosines of lines bisecting the angles between them are (l1 ± l2, m1 ± m2, n1 ± n2).
• The direction cosines of angular bisectors between the lines whose direction cosines are given by (l1, m1, n1) and (l2, m2, n2) 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
cos2 α + cos2β + cos2γ + cos2δ = \frac{4}{3}
sin2 α + sin2β + sin2γ + sin2δ = \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 l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then cos θ = |l1l2 + m1m2 + n1n2|

2. If a1, b1, c1 and a2, b2, c2 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 = |l1l2 + m1m2 + n1n2|.