Advertisements
Advertisements
प्रश्न
The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.
Advertisements
उत्तर
The vector equation of the plane passing through `A(bara)` and perpendicular to `barn` is `barr.barn = bara.barn`.
M(1, 2, 0) is the foot of the perpendicular drawn from the origin to the plane.
Then the plane is passing through M and is perpendicular to OM.
If `barm` is the position vector of M, then `barm = hati + 2hatj`.
Normal to the plane is `barn = bar(OM) = hati + 2hatj`
`barm.barn = (hati + 2hatj) . (hati + 2hatj)`
= 1(1) + 2(2)
= 1 + 4
= 5
∴ The vector equation of the required plane is `barr.(hati + 2hatj)` = 5.
संबंधित प्रश्न
Find the vector equation of the line passing through the point having position vector `-2hat"i" + hat"j" + hat"k" "and parallel to vector" 4hat"i" - hat"j" + 2hat"k"`.
Find the vector equation of line passing through the point having position vector `5hat"i" + 4hat"j" + 3hat"k"` and having direction ratios –3, 4, 2.
Find the vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k" "and perpendicular to vectors" hat"i" + hat"j" + hat"k" and 2hat"i" - hat"j" + hat"k"`.
Find the cartesian equations of the line passing through A(–1, 2, 1) and having direction ratios 2, 3, 1.
Show that the lines given by `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1) and (x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/(4)` intersect. Also, find the coordinates of their point of intersection.
A line passes through (3, –1, 2) and is perpendicular to lines `bar"r" = (hat"i" + hat"j" - hat"k") + lambda(2hat"i" - 2hat"j" + hat"k") and bar"r" = (2hat"i" + hat"j" - 3hat"k") + mu(hat"i" - 2hat"j" + 2hat"k")`. Find its equation.
Find the Cartesian equation of the plane passing through A( -1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
The foot of the perpendicular drawn from the origin to a plane is M(1,0,0). Find the vector equation of the plane.
Find the cartesian equation of the plane `bar"r" = (5hat"i" - 2hat"j" - 3hat"k") + lambda(hat"i" + hat"j" + hat"k") + mu(hat"i" - 2hat"j" + 3hat"k")`.
Find the vector equation of the line which passes through the point (3, 2, 1) and is parallel to the vector `2hat"i" + 2hat"j" - 3hat"k"`.
Find the Cartesian equations of the line which passes through the point (–2, 4, –5) and parallel to the line `(x + 2)/(3) = (y - 3)/(5) = (z + 5)/(6)`.
Find the vector equation of the line which passes through the origin and the point (5, –2, 3).
Find the vector equation of the line which passes through the origin and intersect the line x – 1 = y – 2 = z – 3 at right angle.
If the lines `(x - 1)/(2) = (y + 1)/(3) = (z -1)/(4) and (x- 2)/(1) = (y +m)/(2) = (z - 2)/(1)` intersect each other, find m.
Find the Cartesian equation of the line passing through the origin which is perpendicular to x – 1 = y – 2 = z – 1 and intersect the line `(x - 1)/(2) = (y + 1)/(3) = (z - 1)/(4)`.
The direction ratios of the line which is perpendicular to the two lines `(x - 7)/(2) = (y + 17)/(-3) = (z - 6)/(1) and (x + 5)/(1) = (y + 3)/(2) = (z - 4)/(-2)` are ______.
Find the vector equation of the plane passing through the points A(1, -2, 1), B(2, -1, -3) and C(0, 1, 5).
Solve the following :
A plane makes non zero intercepts a, b, c on the coordinate axes. Show that the vector equation of the plane is `bar"r".(bchat"i" + cahat"j" + abhat"k")` = abc.
Solve the following :
Find the vector equation of the plane passing through the point A(– 2, 3, 5) and parallel to the vectors `4hat"i" + 3hat"k" and hat"i" + hat"j"`.
Solve the following :
Find the cartesian equations of the planes which pass through A(1, 2, 3), B(3, 2, 1) and make equal intercepts on the coordinate axes.
Solve the following :
Find the vector equation of the plane which makes equal non zero intercepts on the coordinate axes and passes through (1, 1, 1).
Solve the following :
Find the vector equation of the plane passing through the origin and containing the line `bar"r" = (hat"i" + 4hat"j" + hat"k") + lambda(hat"i" + 2hat"j" + hat"k")`.
Solve the following :
Show that the lines x = y, z = 0 and x + y = 0, z = 0 intersect each other. Find the vector equation of the plane determined by them.
Find the Cartesian equations of the line passing through A(3, 2, 1) and B(1, 3, 1).
Verify if the point having position vector `4hat"i" - 11hat"j" + 2hat"k"` lies on the line `bar"r" = (6hat"i" - 4hat"j" + 5hat"k") + lambda (2hat"i" + 7hat"j" + 3hat"k")`
Find the vector equation of the line passing through the point having position vector `4hat i - hat j + 2hat"k"` and parallel to the vector `-2hat i - hat j + hat k`.
Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form
Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)
Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2
Find the Cartesian equation of the plane passing through A(7, 8, 6)and parallel to XY plane
Find the Cartesian and vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k"` and perpendicular to vectors `hat"i" + hat"j" + hat"k"` and `2hat"i" - hat"j" + hat"k"`
Find vector equation of the plane passing through A(−2 ,7 ,5) and parallel to vectors `4hat"i" - hat"j" + 3hat"k"` and `hat"i" + hat"j" + hat"k"`
Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes
The cartesian equation of the line `overliner = (hati + hatj + hatk) + lambda(hatj + hatk)` is ______
The line passing through the points (5, 1, a) and (3, b, 1) crosses the YZ – plane at the point `(0, 17/2, (-13)/2)`, then ______.
The centres of the circles x2 + y2 = 1, x2 + y2 + 6x – 2y = 1 and x2 + y2 – 12x + 4y = 1 are ______.
Find the cartesian equation of the plane passing through the point A(–1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
Find the vector equation of the line passing through the points A(2, 3, –1) and B(5, 1, 2).
Show that the lines `(x - 1)/1 = (y - 2)/2 = (z + 1)/-1` and `x/2 = (y - 3)/2 = z/(-1)` do not intersect.
Find the direction cosines of the line `(2x - 1)/3 = 3y = (4z + 3)/2`
