Advertisements
Advertisements
Question
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
Advertisements
Solution
We have, l, m, n and l + δl, m + δm, n + δn, as direction cosines of a variable line of two different positions.
∴ l2 + m2 + n2 = 1 ......(i)
And (l + δ1)2 + (m + δm)2 + (n + δn)2 = 1 ......(ii)
⇒ l2 + m2 + n2 + δl2 + δn2 + 2(lδl + mδm + nδn) = 1
⇒ δl2 + δm2 + δn2 = – 2(lδl + ,m + nδn) .....[∵ l2 + m2 + n2 = 1]
⇒ lδl + mδm + nδn = `(-1)/2` (δl2 + δm2 + δn2) ......(iii)
Now `veca` and `vecb` are unit vectors along a line with direction cosines l, m, n and (l + δl), (m + δm), (n + δn), respectively.
∴ `veca = lhati + mhatj + nhatk` and `vecb = (l + deltal)hati + (m + mdelta)hatj + (n + deltan)hatk`
⇒ cos δθ = (l(l + δl) + m(m + δm) + n(n + δn)
= (l2 + m2 + n2) + (lδl + mδm + nδn)
= `1 - 1/2 (deltal^2 + deltam^2 + deltan^2)` .....[Using equation (iii)]
⇒ 2(1 – cos δθ) = (δ12 + δm2 + δn2)
⇒ `2.2 sin^2 (δtheta)/2` = δ12 + δm2 + δn2
⇒ `4((deltatheta)/2)^2 = deltal^2 + deltam^2 + deltan^2` .....`["Since" (deltatheta)/2 "is small," sin (deltatheta)/2 = (deltatheta)/2]`
⇒ δθ2 + δl2 + δm2 + δn2
APPEARS IN
RELATED QUESTIONS
If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (–4, 3b, –10) and R (8, 14, 2c), then find the values of a, b and c.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(4, –3, 5)
Name the octants in which the following points lie:
(–5, –4, 7)
Name the octants in which the following points lie:
(2, –5, –7)
Planes are drawn parallel to the coordinate planes through the points (3, 0, –1) and (–2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.
The coordinates of a point are (3, –2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.
Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).
If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Are the points A(3, 6, 9), B(10, 20, 30) and C(25, –41, 5), the vertices of a right-angled triangle?
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
Find the co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).
Find the image of the point having position vector `hati + 3hatj + 4hatk` in the plane `hatr * (2hati - hatj + hatk)` + 3 = 0.
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0
Find the foot of perpendicular from the point (2,3,–8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.
Find the equations of the line passing through the point (3,0,1) and parallel to the planes x + 2y = 0 and 3y – z = 0.
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
If the directions cosines of a line are k, k, k, then ______.
The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is ______.
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
The plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The value of α is equal to ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The unit vector normal to the plane x + 2y +3z – 6 = 0 is `1/sqrt(14)hati + 2/sqrt(14)hatj + 3/sqrt(14)hatk`.
