Advertisements
Advertisements
प्रश्न
Find the vector equation of the plane passing through the points A(1, -2, 1), B(2, -1, -3) and C(0, 1, 5).
उत्तर
The vector equation of the plane passing through three non-collinear points `"A"(bara), "B"(barb) and "C"(barc) "is" bar"r".(bar"AB" xx bar"AC") = bar"a".(bar"AB" xx bar"AC")` ...(1)
Here, `bar"a" = hat"i" - 2hat"j" + hat"k", bar"b" = 2hat"i" - hat"j" - 3hat"k", bar"c" = hat"j" + 5hat"k"`
∴ `bar"AB" = bar"b" - bar"a" = (2hat"i" - hat"j" - 3hat"k") - (hat"i" - 2hat"j" + hat"k")`
= `hat"i" + hat"j" - 4hat"k"`
`bar"AC" = bar"c" - bar"a" = (hat"j" + 5hat"k") - (hat"i" - 2hat"j" + hat"k")`
= `hat"i" + 3hat"j" + 4hat"k"`
∴ `bar"AB" xx bar"AC" = |(hati hatj hatk), (1 1-4), (-1 3 4 )|`
= `(4 + 12)hat"i" - (4 - 4)hat"j" + (3 + 1)hat"k"`
= `16hat"i" + 4hat"k"`
Now, `bar"a".(bar"AB" xx bar"AC") = (hat"i" - 2hat"j" + hat"k").(16hat"i" + 4hat"k")`
= (1)(16) + (– 2)(0) + (1)(4) = 20
∴ from(1), the vector equation of the required plane is `bar"r".(16hat"i" + 4hat"k")` = 20.
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 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"`.
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 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 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 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).
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 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 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.
Solve the following :
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 passing through the origin and containing the line `bar"r" = (hat"i" + 4hat"j" + hat"k") + lambda(hat"i" + 2hat"j" + hat"k")`.
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 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
Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes
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 ______
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 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 ______
What is the Cartesian product of A= {l, 2} and B= {a, b}?
Find the direction cosines of the line `(2x - 1)/3 = 3y = (4z + 3)/2`
