A straight line `L` on the xy-plane bisects the angle between `O Xa n dO Ydot` What are the direction cosines of `L ?` a. `<<(1//sqrt(2)),(1//sqrt(2)),0>>` b. `<<(1//2),(sqrt(3)//2),0>>` c. `<<0,0,1>>` d. `<<(2//3),(2//3),(1//3)>>`

Find the angle between the lines whose direction ratios are a,b,c and b-c,c-a,a-b.

Find the angle between the line whose direction cosines are given by l+m+n=0a n d2l^2+2m^2-n^2-0.

If one of the lines of the pair a x^2+2h x y+b y^2=0 bisects the angle between the positive direction of the axes. Then find the relation for a ,b ,h

Find the angle between the lines whose direction cosines are connected by the relations l+m+n=0a n d2lm+2n l-m n=0.

Two straight lines u=0a n dv=0 pass through the origin and the angle between them is tan^(-1)(7/9) . If the ratio of the slope of v=0 and u=0 is 9/2 , then their equations are (a) y+3x=0a n d3y+2x=0 (b) 2y-3x=0a n d3y-x=0 (c) 2y= 3xa n d3y=x (d) y=3xa n d3y=2x

Prove that one of the straight lines given by ax^(2)+2hxy+by^(2)=0 will bisect the angle between the co-ordinate axes if (a+b)^(2)=4h^(2) .

Find the angle between the lines whose direction cosines are given by the equations 3l + m + 5n = 0 and 6mn - 2nl + 5lm = 0

L_1a n dL_2 and two lines whose vector equations are L_1: vec r=lambda((costheta+sqrt(3)) hat i(sqrt(2)sintheta) hat j+(costheta-sqrt(3)) hat k) L_2: vec r=mu(a hat i+b hat j+c hat k) , where lambdaa n dmu are scalars and alpha is the acute angel between L_1a n dL_2dot If the angel alpha is independent of theta, then the value of alpha is a. pi/6 b. pi/4 c. pi/3 d. pi/2

If l_1,m_1,n_1 and l_2,m_2,n_2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_1n_2 - m_2n_1, n_1l_2 -n_2l_1, l_1m_2 -l_2m_1

The base of the pyramid A O B C is an equilateral triangle O B C with each side equal to 4sqrt(2),O is the origin of reference, A O is perpendicualar to the plane of O B C and | vec A O|=2. Then find the cosine of the angle between the skew straight lines, one passing though A and the midpoint of O Ba n d the other passing through O and the mid point of B Cdot