Advertisements
Advertisements
प्रश्न
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
उत्तर
Given, 2l + 2m – n = 0 .....[1]
⇒ n = 2(l + m) ......[2]
And mn + nl + lm = 0
⇒ 2m(l + m) + 2(l + m)l + lm = 0
⇒ 2lm + 2m2 + 2l2 + 2lm + lm = 0
⇒ 2m2 + 5lm + 2l2 = 0
⇒ 2m2 + 4lm + lm + 2l2 = 0
⇒ (2m + l)(m + 2l) = 0
So, we have two cases,
l = – 2m
⇒ – 4m + 2m – n = 0 ......[From 1]
⇒ n = 2m
Hence, direction ratios of one line is proportional to – 2m, m, – 2m or direction ratios are (2, 1, –2)
Another case is,
m = – 2l
⇒ 2l + 2(– 2l) – n = 0
⇒ 2l – 4l = n
⇒ n = – 2l
Hence, direction ratios of another lines is proportional to l, – 2l, – 2l or direction ratios are (1, – 2, – 2)
Therefore, direction vectors of two lines are
`b_1 = -2hati + hatj - 2hatk` and `b_2 = hati - 2hatj - 2hatk`
Also, angle between two lines,
`vecr = veca_1 + lambdavecb_1` and `veca_2 + muvecb_2` is given by
`costheta = |(vecb_1 * vecb_2)/(|vecb_1||vecb_2|)|`
Now, `vecb_1 * vecb_2 = 2(1) + 1(-2) + (-2)(-2)`
= 2 – 2 + 4
= 0
⇒ cos θ = 0
⇒ θ = 90°
Hence, angle between given two lines is 90°
APPEARS IN
संबंधित प्रश्न
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(2, –5, –7)
Find the image of:
(–2, 3, 4) in the yz-plane.
A cube of side 5 has one vertex at the point (1, 0, –1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.
Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Show that the plane ax + by + cz + d = 0 divides the line joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.
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
What is the locus of a point for which y = 0, z = 0?
The length of the perpendicular drawn from the point P(a, b, c) from z-axis is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
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)
Find the co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0
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)`
Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.
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 unit vector normal to the plane x + 2y +3z – 6 = 0 is `1/sqrt(14)hati + 2/sqrt(14)hatj + 3/sqrt(14)hatk`.
The angle between the planes `vecr.(2hati - 3hatj + hatk)` = 1 and `vecr.(hati - hatj)` = 4 is `cos^-1((-5)/sqrt(58))`.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vecr.(5hati - 3hatj - 2hatk)` = 38.
