Advertisements
Advertisements
प्रश्न
`vec"AB" = 3hat"i" - hat"j" + hat"k"` and `vec"CD" = -3hat"i" + 2hat"j" + 4hat"k"` are two vectors. The position vectors of the points A and C are `6hat"i" + 7hat"j" + 4hat"k"` and `-9hat"j" + 2hat"k"`, respectively. Find the position vector of a point P on the line AB and a point Q on the line Cd such that `vec"PQ"` is perpendicular to `vec"AB"` and `vec"CD"` both.
Advertisements
उत्तर
Position vector of A is `6hat"i" + 7hat"j" + 4hat"k"` and `vec"AB" = 3hat"i" - hat"j" + hat"k"`
So, equation of any line passing through A and parallel to `vec"AB"`
`vec"r" = (6hat"i" + 7hat"j" + 4hat"k") + lambda(3hat"i" - hat"j" + hat"k")` .....(i)
Now any point P on `vec"AB" = (6 + 3lambda, 7 - lambda, 4 + lambda)`
Similarly, position vector of C is `-9hat"j" + 2hat"k"`
And `vec"CD" = -3hat"i" + 2hat"j" + 4hat"k"`
So, equation of any line passing through C and parallel to CD is
`vec"r" = (-9hat"j" + 2hat"k") + mu(-3hat"i" + 2hat"j" + 4hat"k")` .....(ii)
Any point Q on `vec"CD" = (-3mu, -9 + 2mu, 2 + 4mu)`
d’ratios of PQ are `(6 + 3lambda + 3mu, 7 - lambda + 9 - 2mu, 4 + lambda - 2 - 4mu)`
⇒ `(6 + 3lambda + 3mu), (16 - lambda - 2mu), (2 + lambda - 4mu)`
Now `vec"PQ"` is ⊥ to equation (i), then
3(6 + 3λ + 3m) – 1(16 – λ – 2m) + 1(2 + λ – 4m) = 0
⇒ 18 + 9λ + 9m – 16 + λ + 2m + 2 + λ – 4m = 0
⇒ 11λ + 7m + 4 = 0 .....(iii)
`vec"PQ"` is also ⊥ to equation (ii), then
`-3(6 + 3lambda + 3mu) + 2(16 - lambda - 2mu) + 4(2 + lambda - 4mu)` = 0
⇒ – 18 – 9λ – 9m + 32 – 2λ – 4m + 8 + 4λ – 16m = 0
⇒ – 7λ – 29m + 22 = 0
⇒ 7λ + 29m – 22 = 0 ......(iv)
Solving equation (iii) and (iv) we get
77λ + 49μ + 28 = 0
77λ + 319μ – 242 = 0
(–) (–) (+)
– 270μ + 270 = 0
∴ μ = 1
Now using μ = 1 in equation (iv) we get
7λ + 29 – 22 = 0
⇒ λ = – 1
∴ Position vector of P = [6 + 3(– 1), 7 + 1, 4 – 1]
= (3, 8, 3)
And position vector of Q = [– 3(1), –9 + 2(1), 2 + 4(1)]
= (– 3, –7, 6)
Hence, the position vectors of P = `3hat"i" + 8hat"j" + 3hat"k"` and Q = `-3hat"i" - 7hat"j" + 6hat"k"`
APPEARS IN
संबंधित प्रश्न
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector.`3hati + 5hatj - 6hatk`
If the points (1, 1, p) and (−3, 0, 1) be equidistant from the plane `vecr.(3hati + 4hatj - 12hatk)+ 13 = 0`, then find the value of p.
If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP.
Find the vector equations of the coordinate planes.
Find the vector equation of each one of following planes.
x + y = 3
The coordinates of the foot of the perpendicular drawn from the origin to a plane are (12, −4, 3). Find the equation of the plane.
Show that the normals to the following pairs of planes are perpendicular to each other.
Find the vector equation of a plane which is at a distance of 3 units from the origin and has \[\hat{k}\] as the unit vector normal to it.
Find the vector equation of the plane passing through the points P (2, 5, −3), Q (−2, −3, 5) and R (5, 3, −3).
Find the vector equation of the plane passing through the points (1, 1, −1), (6, 4, −5) and (−4, −2, 3).
Find the vector equation of the plane passing through the points \[3 \hat{i} + 4 \hat{j} + 2 \hat{k} , 2 \hat{i} - 2 \hat{j} - \hat{k} \text{ and } 7 \hat{i} + 6 \hat{k} .\]
Determine the value of λ for which the following planes are perpendicular to each other.
Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1.
Find the equation of the plane passing through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane x + 2y + 2z = 5.
Find the equation of the plane that contains the point (1, −1, 2) and is perpendicular to each of the planes 2x + 3y − 2z = 5 and x + 2y − 3z = 8.
Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
Find the equation of the plane passing through the intersection of the planes x − 2y + z = 1 and 2x + y + z = 8 and parallel to the line with direction ratios proportional to 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from this plane
Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane 2x + y + z = 7.
Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line \[\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .\]
Hence, or otherwise, deduce the length of the perpendicular.
Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point P (3, 2, 1) from the plane 2x − y + z + 1 = 0. Also, find the image of the point in the plane.
Find the direction cosines of the unit vector perpendicular to the plane \[\vec{r} \cdot \left( 6 \hat{i} - 3 \hat{j} - 2 \hat{k} \right) + 1 = 0\] passing through the origin.
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x − 3y + 4z − 6 = 0.
Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector \[2 \hat{i} + 3 \hat{j} + 4 \hat{k} \] to the plane \[\vec{r} . \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) - 26 = 0\] Also find image of P in the plane.
Find the distance of the point P (–1, –5, –10) from the point of intersection of the line joining the points A (2, –1, 2) and B (5, 3, 4) with the plane x – y + z = 5.
Write the equation of the plane parallel to the YOZ- plane and passing through (−4, 1, 0).
Write the equation of the plane passing through points (a, 0, 0), (0, b, 0) and (0, 0, c).
Write the general equation of a plane parallel to X-axis.
Write the distance between the parallel planes 2x − y + 3z = 4 and 2x − y + 3z = 18.
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.
The vector equation of the plane containing the line \[\vec{r} = \left( - 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + \lambda\left( 3 \hat{i} - 2 \hat{j} - \hat{k} \right)\] and the point \[\hat{i} + 2 \hat{j} + 3 \hat{k} \] is
The equation of the plane parallel to the lines x − 1 = 2y − 5 = 2z and 3x = 4y − 11 = 3z − 4 and passing through the point (2, 3, 3) is
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).
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
Find the equations of the line passing through the point (3, 0, 1) and parallel to the planes x + 2y = 0 and 3y – z = 0.
Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.
The equation of a line, which is parallel to `2hat"i" + hat"j" + 3hat"k"` and which passes through the point (5, –2, 4), is `(x - 5)/2 = (y + 2)/(-1) = (z - 4)/3`.
The coordinates of the foot of the perpendicular drawn from the point A(1, 0, 3) to the join of the points B(4, 7, 1) and C(3, 5, 3) are
