Advertisements
Advertisements
प्रश्न
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
Advertisements
उत्तर
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is 2.
Explanation:
Note that sin2α + sin2β + sin2γ = (1 – cos2α) + (1 – cos2β) + (1 – cos2γ)
= 3 – (cos2α + cos2β + cos2γ)
= 2.
APPEARS IN
संबंधित प्रश्न
If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
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
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
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
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
Write the distance of the point P (x, y, z) from XOY plane.
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,
For every point P (x, y, z) on the x-axis (except the origin),
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 angle between the two diagonals of a cube is
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
Verify whether the following ratios are direction cosines of some vector or not
`4/3, 0, 3/4`
Find the direction cosines of a vector whose direction ratios are
`1/sqrt(2), 1/2, 1/2`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 4hat"j" + 8hat"k"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `3vec"a"- 2vec"b"+ 5vec"c"`
Choose the correct alternative:
The unit vector parallel to the resultant of the vectors `hat"i" + hat"j" - hat"k"` and `hat"i" - 2hat"j" + hat"k"` is
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.
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 direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
Find the direction cosine of a line which makes equal angle with coordinate axes.
What will be the value of 'P' so that the lines `(1 - x)/3 = (7y - 14)/(2P) = (z - 3)/2` and `(7 - 7x)/(3P) = (y - 5)/1 = (6 - z)/5` at right angles.
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 ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
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 ______.
