Advertisements
Advertisements
Question
Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)
Advertisements
Solution
Equation of plane through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0) is
`[(vecr - (2hati + 2hatj + hatk)]*[(hati - 2hatj) xx (hati - hatj - hatk)]` = 0
i.e. `vecr*(2hati + hatj + hatk)` = 7 or 2x + y + z – 7 = 0 ......(1)
Equation of line through (3, – 4, – 5) and (2, – 3, 1) is
`(x - 3)/(-1) = (y + 4)/1 = (z + 5)/6` ......(2)
Any point on line (2) is `(-lambda + 3, lambda - 4, 6lambda - 5)`.
This point lies on plane (1).
Therefore, `2(-lambda + 3) + (lambda - 4) + (6lambda - 5) - 7` = 0
i.e., `lambda` = z
Hence the required point is (1, – 2, 7).
APPEARS IN
RELATED QUESTIONS
The x-axis and y-axis taken together determine a plane known as_______.
Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) and C (–1, 1, 2). Find the coordinates of the fourth vertex.
Name the octants in which the following points lie:
(–5, 4, 3)
Name the octants in which the following points lie:
(7, 4, –3)
Name the octants in which the following points lie:
(–5, –4, 7)
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(–2, 3, 4) in the yz-plane.
Planes are drawn through the points (5, 0, 2) and (3, –2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Write the distance of the point P(3, 4, 5) from z-axis.
The coordinates of the mid-points of sides AB, BC and CA of △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.
Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
Let (3, 4, –1) and (–1, 2, 3) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are
If a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane
Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that
`1/a^2 + 1/b^2 + 1/c^2 = 1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2)`
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is ______.
The locus represented by xy + yz = 0 is ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The cartesian equation of the plane `vecr * (hati + hatj - hatk)` is ______.
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
