The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as l1, m1, n1 ; l2, m2, n2 and l3, m3, n3 are

Question

The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as l1, m1, n1 ; l2, m2, n2 and l3, m3, n3 are (A) l1+l2+l3, m1+m2+m3, n1+n2+n3 (B) (l1+l2+l3/√3), (m1+m2+m3/√3), (n1+n2+n3/√3) (C) (l1+l2+l3/3), (m1+m2+m3/3), (n1+n2+n3/3) (D) none of these

Tardigrade Admin · Accepted Answer

Since the three lines are mutually perpendicular, ∴ l1 l2+m1 m2+n1 n2=0 l2 l3+m2 m3+n2 n3=0 l3 l1+m3 m1+n3 n1=0 Also, l12+m12+n12 =1, l22+m22+n22=1, l32+m32+n32 =1 Now, (l1+l2+l3)2+(m1+m2+m3)2+(n1+n2+n3)2 =(l12+m12+n12)+(l22+m22+n22) +(l32+m32+n32)+2(l1 l2+m1 m2+n1 n2) +2(l2 l3+m2 m3+n2 n3) +2(l3 l1+m3 m1+n3 n1) = 3 ⇒ (l1+l2+l3)2+(m1+m2+m3)2 +(n1+n2+n3)2=3 Here, direction cosines of required line are ((l1+l2+l3/√3), (m1+m2+m3/√3), (n1+n2+n3/√3))

Q. 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

$l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}, n_{1} + n_{2} + n_{3}$

$\frac{l _{1} + l _{2} + l _{3}}{3}, \frac{m _{1} + m _{2} + m _{3}}{3}, \frac{n _{1} + n _{2} + n _{3}}{3}$

$\frac{l _{1} + l _{2} + l _{3}}{3}, \frac{m _{1} + m _{2} + m _{3}}{3}, \frac{n _{1} + n _{2} + n _{3}}{3}$

none of these

Q. The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as l1​,m1​,n1​;l2​,m2​,n2​ and l3​,m3​,n3​ are

l1​+l2​+l3​,m1​+m2​+m3​,n1​+n2​+n3​

3​l1​+l2​+l3​​,3​m1​+m2​+m3​​,3​n1​+n2​+n3​​

3l1​+l2​+l3​​,3m1​+m2​+m3​​,3n1​+n2​+n3​​

none of these

Q. 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

$l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}, n_{1} + n_{2} + n_{3}$

$\frac{l _{1} + l _{2} + l _{3}}{3}, \frac{m _{1} + m _{2} + m _{3}}{3}, \frac{n _{1} + n _{2} + n _{3}}{3}$

$\frac{l _{1} + l _{2} + l _{3}}{3}, \frac{m _{1} + m _{2} + m _{3}}{3}, \frac{n _{1} + n _{2} + n _{3}}{3}$