Advertisements
Advertisements
Question
The equations of motion of a rocket are:
x = 2t,y = –4t, z = 4t, where the time t is given in seconds, and the coordinates of a ‘moving point in km. What is the path of the rocket? At what distances will the rocket be from the starting point O(0, 0, 0) and from the following line in 10 seconds? `vecr = 20hati - 10hatj + 40hatk + μ(10hati - 20hatj + 10hatk)`
Solution
Eliminating t between the equations, we obtain the equation of the path `x/2 = y/(-4) = z/4`, which is the equation of the line passing through the origin having direction ratios <2, –4, 4>. This line is the path of the rocket.
When t = 10 seconds, the rocket will be at the point (20, –40, 40).
Hence, the required distance from the origin at 10 seconds
= `sqrt(20^2 + 40^2 + 40^2)` km
= 20 × 3 km
= 60 km
The distance of the point (20, –40, 40) from the given line
= `(|(veca_2 - veca_1) xx vecb|)/|vecb|`
= `(|-30hatj xx (10hati - 20hatj + 10hatk)|)/(|10hati - 20hatj + 10hatk|)` km
= `(|-300hati + 300hatk|)/(|10hati - 20hatj + 10hatk|)` km
= `(300sqrt(2))/(10sqrt(6))` km
= `10sqrt(3)` km
APPEARS IN
RELATED QUESTIONS
Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`
Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C(5, 3, −3).
Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.
In the given cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(2, 3, – 5) x + 2y – 2z = 9
In the given cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(– 6, 0, 0) 2x – 3y + 6z – 2 = 0
Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr = 2hati -hatj + 2hatk + lambda(3hati + 4hatj + 2hatk)` and the plane `vecr.(hati -hatj + hatk) = 5`.
Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
(A) 2 units
(B) 4 units
(C) 8 units
(D)`2/sqrt29 "units"`
Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0
Write the equation of a plane which is at a distance of \[5\sqrt{3}\] units from origin and the normal to which is equally inclined to coordinate axes.
Show that the points \[\hat{i} - \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 3 \hat{j} + 3 \hat{k} \] are equidistant from the plane \[\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7 \hat{k} \right) + 9 = 0 .\]
Find the equations of the planes parallel to the plane x + 2y − 2z + 8 = 0 that are at a distance of 2 units from the point (2, 1, 1).
Show that the points (1, 1, 1) and (−3, 0, 1) are equidistant from the plane 3x + 4y − 12z + 13 = 0.
Find the distance of the point (3, 3, 3) from the plane \[\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7k \right) + 9 = 0\]
Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6.
If the product of the distances of the point (1, 1, 1) from the origin and the plane x − y + z+ λ = 0 be 5, find the value of λ.
Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C (5, 3, −3).
Find the equation of the plane which passes through the point (3, 4, −1) and is parallel to the plane 2x − 3y + 5z + 7 = 0. Also, find the distance between the two planes.
Find the distance between the planes \[\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + 7 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + 4 \hat{j} + 6 \hat{k} \right) + 7 = 0 .\]
The image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0 is
The distance of the line \[\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - \hat{j}+ 4 \hat{k} \right)\] from the plane \[\vec{r} \cdot \left( \hat{i} + 5 \hat{j} + \hat{k} \right) = 5\] is
If a plane passes through the point (1, 1, 1) and is perpendicular to the line \[\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}\] then its perpendicular distance from the origin is ______.
Find the distance of the point `4hat"i" - 3hat"j" + hat"k"` from the plane `bar"r".(2hat"i" + 3hat"j" - 6hat"k")` = 21.
Find the distance of the point (1, 1 –1) from the plane 3x +4y – 12z + 20 = 0.
Solve the following:
Find the distance of the point `3hat"i" + 3hat"j" + hat"k"` from the plane `bar"r".(2hat"i" + 3hat"j" + 6hat"k")` = 21.
The equation of the plane passing through (3, 1, 2) and making equal intercepts on the coordinate axes is _______.
The equations of planes parallel to the plane x + 2y + 2z + 8 = 0, which are at a distance of 2 units from the point (1, 1, 2) are ________.
If the foot of perpendicular drawn from the origin to the plane is (3, 2, 1), then the equation of plane is ____________.
Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`
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)
A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/ϒ` = 3
Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`
The distance of the plane `vec"r" *(2/7hat"i" + 3/4hat"j" - 6/7hat"k")` = 1 from the origin is ______.
S and S are the focii of the ellipse `x^2/a^2 + y^2/b^2 - 1` whose one of the ends of the minor axis is the point B If ∠SBS' = 90°, then the eccentricity of the ellipse is
A stone is dropped from the top of a cliff 40 m high and at the same instant another stone is shot vertically up from the foot of the cliff with a velocity 20 m per sec. Both stones meet each other after
The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs ₹ 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to ₹ 300 per hour is
`phi` is the angle of the incline when a block of mass m just starts slipping down. The distance covered by the block if thrown up the incline with an initial speed u0 is
Find the distance of the point (2, 3, 4) measured along the line `(x - 4)/3 = (y + 5)/6 = (z + 1)/2` from the plane 3x + 2y + 2z + 5 = 0.
If the distance of the point (1, 1, 1) from the plane x – y + z + λ = 0 is `5/sqrt(3)`, find the value(s) of λ.
Find the distance of the point (1, –2, 0) from the point of the line `vecr = 4hati + 2hatj + 7hatk + λ(3hati + 4hatj + 2hatk)` and the point `vecr.(hati - hatj + hatk)` = 10.
If the points (1, 1, λ) and (–3, 0, 1) are equidistant from the plane `barr*(3hati + 4hatj - 12hatk) + 13` = 0, find the value of λ.
The distance of the point `2hati + hatj - hatk` from the plane `vecr.(hati - 2hatj + 4hatk)` = 9 will be ______.
Find the equations of the planes parallel to the plane x – 2y + 2z – 4 = 0 which is a unit distance from the point (1, 2, 3).
