# If the equations of two lines L1 and L2 are $$\vec{r}=\vec{a_1}+λ\vec{b_1}$$ and $$\vec{r}=\vec{a_2}+μ\vec{b_2}$$, then which of the following is the correct formula for the angle between the two lines?

Category: QuestionsIf the equations of two lines L1 and L2 are $$\vec{r}=\vec{a_1}+λ\vec{b_1}$$ and $$\vec{r}=\vec{a_2}+μ\vec{b_2}$$, then which of the following is the correct formula for the angle between the two lines?
Editor">Editor Staff asked 11 months ago

If the equations of two lines L1 and L2 are \vec{r}=\vec{a_1}+λ\vec{b_1} and \vec{r}=\vec{a_2}+μ\vec{b_2}, then which of the following is the correct formula for the angle between the two lines?

(a) cos⁡θ=\left |\frac{\vec{a_1}.\vec{a_2}}{|\vec{b_1}||\vec{a_2}|}\right |

(b) cos⁡θ=\left |\frac{\vec{a_1}.\vec{a_2}}{|\vec{a_1}||\vec{a_2}|}\right |

(c) cos⁡θ=\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |

(d) cos⁡θ=\left |\frac{\vec{a_1}.\vec{b_2}}{|\vec{a_1}||\vec{b_2}|}\right |

I got this question at a job interview.

Query is from Three Dimensional Geometry topic in section Three Dimensional Geometry of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options
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Editor">Editor Staff answered 11 months ago

The correct option is (c) cos⁡θ=\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |

The best explanation: Given that the equations of the lines are

\vec{r}=\vec{a_1}+λ\vec{b_1} \,and \,\vec{r}=\vec{a_2}+μ\vec{b_2}

∴ the angle between the two lines is given by

cos⁡θ=\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |.