If the direction cosines of two lines are suc | vedclass

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

Question

(B) Given equations for direction cosines $(l, m, n)$ are:$l+m+n=0$ $\dots(i)$$l^2+m^2-n^2=0$ $\dots(ii)$Also, we know that $l^2+m^2+n^2=1$ $\dots(iii

Vedclass · Accepted Answer

$\pi / 3$

Vedclass · Answer

(B) Given equations for direction cosines $(l, m, n)$ are:$l+m+n=0$ $\dots(i)$$l^2+m^2-n^2=0$ $\dots(ii)$Also, we know that $l^2+m^2+n^2=1$ $\dots(iii)$From $(i)$, $l+m = -n$. Squaring both sides, $l^2+m^2+2lm = n^2$, so $l^2+m^2 = n^2-2lm$.Substituting this into $(ii)$, we get $(n^2-2lm) - n^2 = 0$, which implies $-2lm = 0$, so $l=0$ or $m=0$.Case $1$: If $l=0$, then from $(i)$, $m+n=0 \implies m=-n$. Substituting into $(iii)$, $0^2+(-n)^2+n^2=1 \implies 2n^2=1 \implies n = \pm 1/\sqrt{2}$.Thus, the direction cosines are $(0, 1/\sqrt{2}, -1/\sqrt{2})$ and $(0, -1/\sqrt{2}, 1/\sqrt{2})$.Case $2$: If $m=0$, then from $(i)$, $l+n=0 \implies l=-n$. Substituting into $(iii)$, $(-n)^2+0^2+n^2=1 \implies 2n^2=1 \implies n = \pm 1/\sqrt{2}$.Thus, the direction cosines are $(1/\sqrt{2}, 0, -1/\sqrt{2})$ and $(-1/\sqrt{2}, 0, 1/\sqrt{2})$.Let the two lines have direction cosines $(l_1, m_1, n_1) = (0, 1/\sqrt{2}, -1/\sqrt{2})$ and $(l_2, m_2, n_2) = (1/\sqrt{2}, 0, -1/\sqrt{2})$.The cosine of the angle $	heta$ between them is given by $\cos 	heta = |l_1l_2 + m_1m_2 + n_1n_2| = |0(1/\sqrt{2}) + (1/\sqrt{2})(0) + (-1/\sqrt{2})(-1/\sqrt{2})| = |1/2| = 1/2$.Therefore, $	heta = \cos^{-1}(1/2) = \pi/3$.

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:

Similar Questions

If the direction cosines of two lines are such that $2l + m + 2n = 0$ and $3l^2 + 5m^2 - 11n^2 = 0$,then the angle between the two lines is

Find a vector $\vec{r}$ of magnitude $3 \sqrt{2}$ units which makes an angle of $\frac{\pi}{4}$ and $\frac{\pi}{2}$ with $y$ and $z$-axes,respectively.

If the projections of a line segment on the $x, y$ and $z-$ axes in $3-$ dimensional space are $2, 3$ and $6$ respectively,then the length of the line segment is

The projections of a line on the coordinate axes are $4, 6, 12$. The direction cosines of the line are

If a line makes an angle of $\frac{\pi}{3}$ with each $X$ and $Y$ axis,then the acute angle made by $Z$-axis is

Vedclass Test Series

Exam Paper Generator

Online Exam Module