Advertisements
Advertisements
प्रश्न
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"`.
Advertisements
उत्तर
If `hat"n"` is a unit vector along the normal and p i the length of the perpendicular from origin to the plane, then the vector equation of the plane `bar"r".hat"n" = p`
Here, `bar"n" = 2hat"i" + hat"j" + 2hat"k"` and p = 5
∴ `|bar"n"| = sqrt(2^2 + 1^2 + (2)^2`
= `sqrt(9)`
= 3
`hat"n" = bar"n"/|bar"n"|`
= `(1)/(3)(2hat"i" + hat"j" + 2hat"k")`
∴ the vector equation of the required plane is
`bar"r".[1/3(2hat"i" + hat"j" + 2hat"k")]` = 5
i.e. `bar"r".(2hat"i" + hat"j" + 2hat"k")` = 15.
APPEARS IN
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 the line passing through points having position vector `3hati + 4hatj - 7hatk and 6hati - hatj + hatk`.
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 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") + λ(3hat"i" + 2hat"j" + hat"k").`
Find the cartesian equations of the line passing through A(–1, 2, 1) and having direction ratios 2, 3, 1.
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.
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.
Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.
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 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 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 vector equation of the line whose Cartesian equations are y = 2 and 4x – 3z + 5 = 0.
Choose correct alternatives:
The vector equation of line 2x – 1 = 3y + 2 = z – 2 is ______.
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 :
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).
Find the cartesian equation of the plane passing through A(1, 2, 3) and the direction ratios of whose normal are 3, 2, 5.
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 Cartesian equation of the line passing through A(1, 2, 3) and having direction ratios 2, 3, 7
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 plane passing through the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)
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
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 ____________.
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 ______
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 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 ______.
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}?
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 ______.
