The direction cosines of a line equally incli | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the direction cosines of the required line be $(l, m, n)$.Since the line is equally inclined to the three mutually perpendicular lines with di

Vedclass · Accepted Answer

$\frac{l_1 + l_2 + l_3}{\sqrt{3}}, \frac{m_1 + m_2 + m_3}{\sqrt{3}}, \frac{n_1 + n_2 + n_3}{\sqrt{3}}$

Vedclass · Answer

(B) Let the direction cosines of the required line be $(l, m, n)$.Since the line is equally inclined to the three mutually perpendicular lines with direction cosines $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$,the cosine of the angle $	heta$ between the required line and each of these lines is the same.Let $\cos 	heta = k$.Then,$l l_1 + m m_1 + n n_1 = k$,$l l_2 + m m_2 + n n_2 = k$,and $l l_3 + m m_3 + n n_3 = k$.Squaring and adding these equations,we get:$(l l_1 + m m_1 + n n_1)^2 + (l l_2 + m m_2 + n n_2)^2 + (l l_3 + m m_3 + n n_3)^2 = 3k^2$.Using the property of orthogonal matrices,for any vector $(l, m, n)$,$\sum (l l_i + m m_i + n n_i)^2 = l^2 + m^2 + n^2 = 1$.Thus,$3k^2 = 1 \implies k = \frac{1}{\sqrt{3}}$.Now,consider the vector $\vec{v} = (l_1+l_2+l_3, m_1+m_2+m_3, n_1+n_2+n_3)$.The magnitude of this vector is $\sqrt{(l_1+l_2+l_3)^2 + (m_1+m_2+m_3)^2 + (n_1+n_2+n_3)^2}$.Since the lines are mutually perpendicular,the sum of squares simplifies to $1+1+1 = 3$.Thus,the unit vector along this direction is $\left( \frac{l_1+l_2+l_3}{\sqrt{3}}, \frac{m_1+m_2+m_3}{\sqrt{3}}, \frac{n_1+n_2+n_3}{\sqrt{3}} ight)$.

The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$ and $(l_3, m_3, n_3)$ are:

Similar Questions

The direction cosines of the line which bisects the angle between the positive direction of $Y$ and $Z$ axes are

If a unit vector $\vec{r}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$,$\frac{\pi}{4}$ with $\hat{j}$ and $\theta \in (0, \pi)$ with $\hat{k}$,then a value of $\theta$ is

The distance between two points $P$ and $Q$ is $d$ and the lengths of the projections of $PQ$ on the coordinate planes are $d_1, d_2, d_3$. Then $d_1^2 + d_2^2 + d_3^2 = k d^2$,where $k$ is:

The angle between the lines whose direction ratios satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is

If the direction cosines of two lines are such that $l+m+n=0$ and $l^2+m^2-n^2=0$, then the angle between them is

Vedclass Test Series

Exam Paper Generator

Online Exam Module