Advertisements
Advertisements
प्रश्न
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"`.
Advertisements
उत्तर
The vector equation of the plane passing through the point A`(bara)` and parallel to the vectors `bar"b" and bar"c"` is
`bar"r".(bar"b" xx bar"c") = bar"a".(bar"b" xx bar"c")` ...(1)
Here, `bar"a" = -2hat"i" + 3hat"j" + 5hat"k"`
`bar"b" = 4hat"i" + 3hat"k"`,
`bar"c" = hat"i" + hat"j"`
∴ `bar"b" xx bar"c" = |(hat"i", hat"j", hat"k"),(4, 0, 3),(1, 1, 0)|`
= `(0 - 3)hat"i" - (0 - 3)hat"j" - (4 - 0)hat"k"`
= `-3hat"i" + 3hat"j" + 4hat"k"`
∴ `bar"r".(bar"b" xx bar"c") = bar"a".(bar"b" xx bar"c")`
`bar"r". (-3hat"i" + 3hat"j" + 4hat"k") = (-2hat"i" + 3hat"j" + 5hat"k").(-3hat"i" + 3hat"j" + 4hat"k")`
= 6 + 9 + 20
= 35
∴ From (1), the vector equation of the required plane is `bar"r".(- 3hat"i" + 3hat"j" + 4hat"k")` = 35.
APPEARS IN
संबंधित प्रश्न
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).
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(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 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 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)`.
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 Cartesian equations of the line passing through the point A(1, 1, 2) and perpendicular to the vectors `barb = hati + 2hatj + hatk and barc = 3hati + 2hatj - hatk`.
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.
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 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).
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 cartesian equation of the plane `bar"r" = lambda(hat"i" + hat"j" - hat"k") + mu(hat"i" + 2hat"j" + 3hat"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 equation of the line passing through A(1, 2, 3) and B(2, 3, 4)
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 A(7, 8, 6)and parallel to XY plane
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 cartesian coordinates of the point on the parabola y2 = x whose parameter 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 ______.
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 ______.
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.
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 ______.
Find the direction cosines of the line `(2x - 1)/3 = 3y = (4z + 3)/2`
