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.
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 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(2, 2, 1) and B(1, 3, 0).
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(7, 8, 6) and parallel to the XY plane.
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 plane which makes intercepts 1, 1, 1 on the co-ordinates axes.
Find the vector equation of the line passing through the point having position vector `3hat"i" + 4hat"j" - 7hat"k"` and parallel to `6hat"i" - hat"j" + hat"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 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)`.
Find the vector and Cartesian equations of the line passing through the point (–1, –1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z − 2.
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)`.
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 ______.
Solve the following :
Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector `2hat"i" + hat"j" + 2hat"k"`.
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 vector equation of the plane which makes equal non zero intercepts on the coordinate axes and passes through (1, 1, 1).
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`
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`.
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 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 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"`
The vector equation of the line passing through `4hati - hatj + 2hatk` and parallel to `-2hati - 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 lines x = ay + b, z = cy + d and x = a'y + b', z = c'y + d' are perpendicular to each other, if ______
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 vector equation of the line passing through the points A(2, 3, –1) and B(5, 1, 2).
Find the vector equation of a line passing through the point `hati + 2hatj + 3hatk` and perpendicular to the vectors `hati + hatj + hatk` and `2hati - hatj + hatk`.
