Advertisements
Advertisements
Question
Find the vector equation of the plane which makes intercepts 1, 1, 1 on the co-ordinates axes.
Advertisements
Solution
The vector equation of the plane passing through `"A"(bara), "B"(barb), "C"(barc)`, where A, B, C are non-collinear is `bar"r".(bar"AB" xx bar"AC") = bar"a".(bar"AB" xx bar"AC")` ...(1)
The required plane makes intercepts 1, 1, 1 on the coordinate axes.
∴ It passes through the three non-collinear points
A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1)
∴ `bar"a" = hat"i", bar"b" = hat"j", bar"c" = hat"k"`
`bar"AB" = bar"b" - bar"a" = hat"j" - hat"i" = -hat"i" + hat"j"`
∴ `bar"AC" = bar"c" - bar"a" = hat"k" - hat"i" = -hat"i" + hat"k"`
∴ `bar"AB" xx bar"AC" = |(hat"i", hat"j", hat"k"),(-1, 1, 0),(-1, 0, 1)|`
= `(1 - 0)hat"i" - (- 1 + 0)hat"j" + (0 + 1)hat"k"`
= `hat"i" + hat"j" + hat"k"`
Also, `bar"a".(bar"AB" xx bar"AC")` = `hat"i".(hat"i" + hat"j" + hat"k")`
= 1 × 1 + 0 × 1 + 0 × 1
= 1
∴ From (1), the vector equation of the required plane is `bar"r".(hat"i" + hat"j" + hat"k")` = 1.
APPEARS IN
RELATED QUESTIONS
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.
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 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 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 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(1,-2, 3) and direction ratios of whose normal are 0, 2, 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 :
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"`.
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")`.
Find the vector equation of the line `x/1 = (y - 1)/2 = (z - 2)/3`
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 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
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"`
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 cartesian equation of the line `overliner = (hati + hatj + hatk) + lambda(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 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 ______
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 ______.
What is the Cartesian product of A= {l, 2} and B= {a, b}?
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.
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`.
Find the direction cosines of the line `(2x - 1)/3 = 3y = (4z + 3)/2`
If vector equation of the line `(x - 2)/2 = (2y - 5)/-3 = z + 1 "is" barr = (2hati + 5/2 hatj - hatk) + lambda (2hati - 3/2 hatj + phatk)` then p is equal to ______.
