Advertisements
Advertisements
Question
The foot of the perpendicular drawn from the origin to a plane is M(1,0,0). Find the vector equation of the plane.
Advertisements
Solution
The vector equation of the plane passing through A`(bara)` and perpendicular to `bar"n" "is" bar"r".bar"n" = bar"a".bar"n"`.
M(1,0,0) is the foot of the perpendicular drawn from origin to the plane. Then the plane is passing through M and is perpendicular to OM.
If `bar"m"` is the position vector of M, then `bar"m" = hat"i"`.
Normal to the plane is
`bar"n" = bar"OM" = hat"i"`
`bar"m".bar"n" = hat"i".hat"i"` = 1
∴ the vector equation of the required plane is
`bar"r".hat"i"` = 1.
APPEARS IN
RELATED QUESTIONS
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"`.
A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.
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.
Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.
Find the Cartesian equation of the plane passing through A( -1, 2, 3), the direction ratios of whose normal are 0, 2, 5.
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"`.
Obtain the vector equation of the line `(x + 5)/(3) = (y + 4)/(5)= (z + 5)/(6)`.
Find the Cartesian equations of the line which passes through points (3, –2, –5) and (3, –2, 6).
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)`.
Find the coordinates of points on th line `(x - 1)/(1) = (y - 2)/(-2) = (z - 3)/(2)` which are at the distance 3 unit from the base point A(l, 2, 3).
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 :
Find the cartesian equation of the plane passing through A(1,-2, 3) and direction ratios of whose normal are 0, 2, 0.
Solve the following :
Find the cartesian equation of the plane passing through A(7, 8, 6) and parallel to the plane `bar"r".(6hat"i" + 8hat"j" + 7hat"k")` = 0.
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 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).
Find the vector equation of the line `x/1 = (y - 1)/2 = (z - 2)/3`
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`.
Find the Cartesian equation of the plane passing through the points (3, 2, 1) and (1, 3, 1)
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 Cartesian equation of the line passing through the point A(2, 1, −3) and perpendicular to vectors `hat"i" + hat"j" + hat"k"` and `hat"i" + 2hat"j" - hat"k"`
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 the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)
The cartesian equation of the line `overliner = (hati + hatj + hatk) + lambda(hatj + hatk)` is ______
If the line passes through the points P(6, -1, 2), Q(8, -7, 2λ) and R(5, 2, 4) then value of λ is ______.
The shortest distance between A (1, 0, 2) and the line `(x + 1)/3 = (y - 2)/(-2) = (z + 1)/(-1)` is given by line joining A and B, then B in the line is ______
The lines x = ay + b, z = cy + d and x = a'y + b', z = c'y + d' are perpendicular to each other, if ______
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 ______.
A line passes through the point of intersection of the lines 3x + y + 1 = 0 and 2x – y + 3 = 0 and makes equal intercepts with axes. The equation of the line is ______.
The centres of the circles x2 + y2 = 1, x2 + y2 + 6x – 2y = 1 and x2 + y2 – 12x + 4y = 1 are ______.
What is the Cartesian product of A= {l, 2} and B= {a, b}?
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.
Show that the lines `(x - 1)/1 = (y - 2)/2 = (z + 1)/-1` and `x/2 = (y - 3)/2 = z/(-1)` do not intersect.
If the line `(x - 1)/2 = (y + 1)/3 = z/4` lies in the plane 4x + 4y – kz = 0, then the value of k is ______.
