Advertisements
Advertisements
प्रश्न
Solve the following :
Find the vector equation of the plane which bisects the segment joining A(2, 3, 6) and B(4, 3, –2) at right angle.
Advertisements
उत्तर
The vector equation of the plane passing through `"A"(bara)` and perpendicular to the vector `bar"n"` is `bar"r".bar"n" = bar"a".bar"n"` ...(1)
The position vectors `bar"a" and bar"b"` of the given points A and B are `bar"a" = 2hat"i" + 3hat"j" + 6hat"k" and bar"b" = 4hat"i" + 3hat"j" - 2hat"k"`
If M is the midpoint of segment AB, the position vector `bar"m"` of M is given by
`bar"m" = (bar"a" + bar"b")/(2)`
= `((2hat"i" + 3hat"j" + 6hat"k") + (4hat"i" + 3hat"j" - 2hat"k"))/(2)`
= `(6hat"i" + 6hat"j" + 4hat"k")/(2)`
= `3hat"i" + 3hat"j" + 2hat"k"`
The plane passes through `"M"(bar"m")`.
AB is perpendicular to the plane
If `bar"n"` is normal to the plane, then `bar"n" = bar"AB"`
∴ `bar"n" = bar"b" - bar"a" = (4hat"i" + 3hat"j" - 2hat"k") - (2hat"i" + 3hat"j" + 6hat"k")`
= `2hat"i" - 8hat"k"`
∴ `bar"m".bar"n" = (3hat"i" + 3hat"j" + 2hat"k").(2hat"i" - 8hat"k")`
= (3)(2) + (3)(0) + (2)(– 8)
= 6 + 0 – 16
= – 10
∴ from (1), the vector equation of the required plane is `bar"r".bar"n" = bar"m".bar"n"`
i.e. `bar"r".(2hat"i" - 8hat"k")` = – 10
i.e. `bar"r".(hat"i" - 4hat"k")` = – 5.
APPEARS IN
संबंधित प्रश्न
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 Cartesian equations of the line passing through A(2, 2, 1) and B(1, 3, 0).
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.
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 passing through A(7, 8, 6) and parallel to the XY plane.
Find the vector equation of the plane which makes intercepts 1, 1, 1 on the co-ordinates axes.
Find the vector equation of the line which passes through the origin and the point (5, –2, 3).
Find the Cartesian equations of the line which passes through the point (2, 1, 3) and perpendicular to the lines `(x - 1)/(1) = (y - 2)/(2) = (z - 3)/(3) and x/(-3) = y/(2) = z/(5)`.
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.
Choose correct alternatives:
The vector equation of line 2x – 1 = 3y + 2 = z – 2 is ______.
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(7, 8, 6) and parallel to the plane `bar"r".(6hat"i" + 8hat"j" + 7hat"k")` = 0.
The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.
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 which makes equal non zero intercepts on the coordinate axes and passes through (1, 1, 1).
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 vector equation of the line `x/1 = (y - 1)/2 = (z - 2)/3`
Find the Cartesian equation of the line passing through A(1, 2, 3) and having direction ratios 2, 3, 7
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 vector equation of the line passing through the point having position vector `-hat"i"- hat"j" + 2hat"k"` and parallel to the line `bar"r" = (hat"i" + 2hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" + hat"k")`, µ is a parameter
Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2
Find m, if the lines `(1 - x)/3 =(7y - 14)/(2"m") = (z - 3)/2` and `(7 - 7x)/(3"m") = (y - 5)/1 = (6 - z)/5` are at right angles
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 point P lies on line A, B where A = (2, 4, 5} and B = (1, 2, 3). If z co-ordinate of point P is 3, the its y co-ordinate is ______.
The cartesian coordinates of the point on the parabola y2 = x whose parameter is ____________.
The vector equation of the line passing through `4hati - hatj + 2hatk` and parallel to `-2hati - hatj + hatk` is ______
The cartesian equation of the line `overliner = (hati + hatj + hatk) + lambda(hatj + hatk)` is ______
If line joining points A and B having position vectors `6overlinea - 4overlineb + 4overlinec` and `-4overlinec` respectively, and the line joining the points C and D having position vectors `-overlinea - 2overlineb - 3overlinec` and `overlinea + 2overlineb - 5overlinec` intersect, then their point of intersection 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 equation of line is `(x - 1)/2 = (y + 1)/(-2) = (z + 1)/1`. The co-ordinates of the point on the line at a distance of 3 units from the point (1, -1, -1) is ______
The equation of line equally inclined to co-ordinate axes and passing through (–3, 2, –5) is ______.
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 ______.
