Advertisements
Advertisements
प्रश्न
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
उत्तर
Given that l, m, n and l + δl, m + δm, n + δn, are the direction cosines of a variable line in two positions
∴ l2 + m2 + n2 = 1 ......(i)
And (l + δl)2 + (m + δm)2 + (n + δn)2 = 1 ......(ii)
⇒ l2 + δl2 + 2l.δl + m2 + δm2 + 2m.δm + n2 + δn2 + 2n.δn = 1
⇒ (l2 + m2 + n2) + (δl2 + δm2 + δn2) + 2(l.δl + m.δm + n.δn) = 1
⇒ 1 + (δl2 + δm2 + δn2) + 2(l.δl + m.δm + n.δn) = 1
⇒ l.δl + m.δm + n.δn =`-1/2(δl^2 + δm^2 + δn^2)`
Let `vec"a"` and `vec"b"` be the unit vectors along a line with d’cosines l, m, n and d (l + δl), (m + δm), (n + δn).
∴ `vec"a" = lhat"i" + mhat"j" + nhat"k"` and `vec"b" = (l + δl)hat"i" + (m + δm)hat"j" + (n + δn)hat"k"`
`cosδtheta = (vec"a"*vec"b")/(|vec"a"||vec"b"|)`
`cosδtheta = ((lhat"i" + mhat"j" + nhat"k").[(l + δl)hat"i" + (m + δm)hat"j" + (n + δn)hat"k"])/(1.1)` .....`[because |vec"a"| = |vec"b"| = 1]`
⇒ cos δθ = l(l + δl) + m(m + δm) + n(n + δn)
⇒ cos δθ = l2 + l.δl + m2 + m.δm + n2 + n.δn
⇒ cos δθ = (l2 + m2 + n2) + (l.δl + m.δm + n.δn)
⇒ cos δθ = `1 - 1/2(δl^2 + δm^2 + δn^2)`
⇒ `1 - cosδtheta = 1/2 (δl^2 + δm^2 + δn^2)`
⇒ `2sin^2 (δtheta)/2 = 1/2 (δ1^2 + δm^2 + δn^2)`
⇒ `4sin^2 (δtheta)/2 = δl^2 + δm^2 + δn^2`
⇒ `4((δtheta)/2)^2 = δl^2 + δm^2 + δn^2` ......`[(because (δtheta)/2 "is very small so"","),(sin (δtheta)/2 = (δtheta)/2)]`
⇒ `(δtheta)^2 = δl^2 + δm^2 + δn^2`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Find the direction cosines of a line which makes equal angles with the coordinate axes.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines
If a line has direction ratios 2, −1, −2, determine its direction cosines.
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.
Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l2 + m2 − n2 = 0
Find the angle between the lines whose direction cosines are given by the equations
2l + 2m − n = 0, mn + ln + lm = 0
Define direction cosines of a directed line.
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.
For every point P (x, y, z) on the xy-plane,
A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is
A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
The distance of the point P (a, b, c) from the x-axis is
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
Find the direction cosines and direction ratios for the following vector
`5hat"i" - 3hat"j" - 48hat"k"`
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
If a line makes an angle of `pi/4` with each of y and z-axis, then the angle which it makes with x-axis is ______.
Find the direction cosine of a line which makes equal angle with coordinate axes.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
The co-ordinates of the point where the line joining the points (2, –3, 1), (3, –4, –5) cuts the plane 2x + y + z = 7 are ______.
A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the line so directed that the angle made by it with positive direction of x-axis is acute, are ______.
If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.
If a line makes an angle α, β and γ with positive direction of the coordinate axes, then the value of sin2α + sin2β + sin2γ will be ______.
