Advertisements
Advertisements
प्रश्न
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Advertisements
उत्तर
\[ \text{ We know that if a line has direction ratio } \left( a, b, c \right), \text{ then the cosine of its angle with the z - axis is given } by\]
\[\cos \gamma = \frac{c}{\sqrt{a^2 + b^2 + c^2}}\]
\[\text { Suppose the inclination of the line with direction ratio } \left( 0, 1, - 1 \right) \text{ with z - axis is } \gamma . \]
\[\text{ Now }, \]
\[\cos \lambda = \frac{- 1}{\sqrt{0 + 1 + 1}}\]
\[ = - \frac{1}{\sqrt{2}} \]
\[ \Rightarrow \lambda = \frac{3\pi}{4}\]
\[\text{ Hence, the inclination of the line with z - axis is } \frac{3\pi}{4} . \]
APPEARS IN
संबंधित प्रश्न
Which of the following represents direction cosines of the line :
(a)`0,1/sqrt2,1/2`
(b)`0,-sqrt3/2,1/sqrt2`
(c)`0,sqrt3/2,1/2`
(d)`1/2,1/2,1/2`
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1, n1l2 − n2l1, l1m2 − l2m1.
If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.
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.
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 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.
What are the direction cosines of Y-axis?
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.
Write the coordinates of the projection of point P (x, y, z) on XOZ-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.
The distance of the point P (a, b, c) from the x-axis is
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are
If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to
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
`5hat"i" - 3hat"j" - 48hat"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 `vec"a", vec"b", vec"c"`
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
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
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 angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`
The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.
If two straight lines whose direction cosines are given by the relations l + m – n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively, is ______.
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 ______.
A directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively. What are these angles called?
Which proportion correctly expresses the relationship between direction ratios \[(a,b,c)\] and direction cosines \[(l,m,n)\]?
For direction ratios \[(a,b,c)\], which set can represent the corresponding direction cosines when the direction of the line is chosen?
In the relation between direction ratios and direction cosines, what does the sign \[\pm\] in \[l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}}\] depend on?
