Advertisements
Advertisements
प्रश्न
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")`.
Advertisements
उत्तर
The equation `bar"r" = bar"a" + lambdabar"b" + mubar"c"` represents a plane passing through a point having position vector `bar"a"` and parallel to vectors `bar"b" and bar"c"`.
Here,
`bar"a" = 5hat"i" - 2hat"j" - 3hat"k",`
`bar"b" = hat"i" + hat"j" + hat"k"`,
`bar"c" = hat"i" - 2hat"j" + 3hat"k"`
∴ `bar"b" xx bar"c" = |(hat"i", hat"j", hat"k"),(1, 1, 1),(1, -2, 3)|`
= `(3 + 2)hat"i" - (3 - 1)hat"j" + (-2 - 1)hat"k"`
= `5hat"i" - 2hat"j" - 3hat"k"`
= `bar"a"`
Also,
`bar"a".(bar"b" xx bar"c") = bar"a".bar"a" = |bar"a"|^2`
= (5)2 + (– 2)2 + (3)2
= 38
The vector equation of the plane passing through A`(bara)` and parallel to `bar"b" and bar"c"` is
`bar"r".(bar"b" xx bar"c") = bar"a".(bar"b" xx bar"c")`
∴ The vector equation of the given plane is
`bar"r".(5hat"i" - 2hat"j" - 3hat"k")` = 38
If `bar"r" = xhat"i" + yhat"j" + zhat"k"`, then this equation becomes
`(xhat"i" + yhat"j" + zhat"k").(5hat"i" - 2hat"j" - 3hat"k")` = 38
∴ 5x – 2y – 3z = 38.
This is the cartesian equation of the required plane.
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 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 vector equation of the plane passing through the point A(– 2, 7, 5) and parallel to vector `4hat"i" - hat"j" + 3hat"k" and hat"i" + hat"j" + hat"k"`.
Find the vector equation of the plane which makes intercepts 1, 1, 1 on the co-ordinates axes.
Obtain the vector equation of the line `(x + 5)/(3) = (y + 4)/(5)= (z + 5)/(6)`.
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 points (3, –2, –5) and (3, –2, 6).
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 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 vector equation of the line whose Cartesian equations are y = 2 and 4x – 3z + 5 = 0.
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).
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 :
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 cartesian equation of the plane `bar"r" = lambda(hat"i" + hat"j" - hat"k") + mu(hat"i" + 2hat"j" + 3hat"k")`.
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 :
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`
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)
Find the direction ratios of the line perpendicular to the lines
`(x - 7)/2 = (y + 7)/(-3) = (z - 6)/1` and `(x + 5)/1 = (y + 3)/2 = (z - 6)/(-2)`
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 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"`
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 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 ______
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 ______.
Equation of Z-axis 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 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}?
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 ______.
