Advertisements
Advertisements
प्रश्न
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
Advertisements
उत्तर
We have A(a, b, c) and O(0, 0, 0)
∴ Direction ratios of OA = a – 0, b – 0, c – 0
∴ Direction cosines of line OA = `"a"/sqrt("a"^2 + "b"^2 + "c"^2)`
`"b"/sqrt("a"^2 + "b"^2 + "c"^2)`
`"c"/sqrt("a"^2 + "b"^2 + "c"^2)`
Now direction ratios of the normal to the plane are (a, b, c).
∴ Equation of the plane passing through the point A(a, b, c) is a(x – a) + b(y – b) + c(z – c) = 0
⇒ ax – a2 + by – b2 + cz – c2 = 0
⇒ ax + by + cz = a2 + b2 + c2
Hence, the required equation is ax + by + cz = a2 + b2 + c2.
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; z=-1`
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
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.
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
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).
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 ratios are proportional to a, b, c and b − c, c − a, a− b.
If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.
Find the angle between the lines whose direction cosines are given by the equations
2l − m + 2n = 0 and mn + nl + lm = 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 distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
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 equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Find the direction cosines of a vector whose direction ratios are
`1/sqrt(2), 1/2, 1/2`
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"`
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.
If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
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 ______.
If the directions cosines of a line are k,k,k, then ______.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
The line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.
Find the direction cosine of a line which makes equal angle with coordinate axes.
The d.c's of a line whose direction ratios are 2, 3, –6, are ______.
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 ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line `(x - 15)/3 = (y - 29)/8 = (z - 5)/-5`.
