Advertisements
Advertisements
प्रश्न
Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.
Advertisements
उत्तर
step 1
The given points are A(2, 2, –1), B(3, 4, 2) and C(7, 0, 6)
Let `vec"a" = 2hat"i" + 2hat"j" - hat"k"`
`vec"b" = 3hat"i" + 4hat"j" + 2hat"k"`
`vec"c" = 7hat"i" + 6hat"k"`
Hence the vector equation of the plane passing through the points
`(vec"r"- vec"a").(vec"AB" xx vec"AC") = 0`
= `(vec"r" - vec"a").((vec"b" - vec"a") xx (vec"c" - vec"a")) = 0`
Now
`vec"b"- vec"a" = (3hat"i"+4hat"j" + 2hat"k")-(2hat"i"+ 2hat"j"-hat"k")`
⇒ `hat"i" + 2hat"j" + 3hat"k"`
`vec"c" - vec"a" = (7hat"i" + 6hat"k") - (2hat"i" + 2hat"j" - hat"k")`
= `5hat"i" - 2hat"j" + 7hat"k"`
So the required vector equation of plane is
`[vec"r" - (2hat"i" + 2hat"j" - hat"k")].[(hat"i" + 2hat"j" + 3hat"k") xx (5hat"i" - 2hat"j" + 7hat"k")] = 0`
Step 2
`(vec"b" - vec"a") xx (vec"c" - vec"a") = |(hat"i",hat"j",hat"k"),(1,2,3),(5,-2,7)|`
= `hat"i" (14 +6) -hat"j" (7 -15) + hat"k" (-2-10)`
= `20hat"i" + 8hat"j" - 12hat"k"`
⇒ `(vec"r" - (2hat"i" + 2hat"j" - hat"k")) . (20hat"i" + 8hat"j"-12hat"k") = 0`
`(vec"r" - (2hat"i" + 2hat"j" - hat"k")) . (5hat"i" + 2hat"j"- 3hat"k") = 0`
`vecr . (5hat"i" + 2hat"j"- 3hat"k") = (2hat"i" + 2hat"j" - hat"k"). (5hat"i" + 2hat"j", 3hat"k")`
`vecr . (5hat"i" + 2hat"j"- 3hat"k") = 10 + 4 + 3`
`vecr . (5hat"i" + 2hat"j"-3hat"k") = 17`
This is the required vector equation of the plane
Step 3
The Cartesian Equation of the plane passing through the three points is given as below-
5x + 2y − 3z − 17 = 0
This is the required cartesian equation of the plane.
The equation of plane parallel to 5x + 2y − 3z − 17 = 0 will be 5x + 2y − 3z + λ = 0
∴ it passes through (4, 3, 1).
So, 5 × 4 + 2 × 3 − 3 × 1 + λ = 0
20 + 6 − 3 + λ = 0
So, λ = −23
so the equation of the plane will be
5x + 2y − 3z − 23 = 0
5x + 2y − 3z = 23
so the vector form of the equation of plane will be
`vecr. (5hat"i" + 2hat"j" - 3hat"k") = 23`
संबंधित प्रश्न
Find the vector equation of the plane passing through a point having position vector `3 hat i- 2 hat j + hat k` and perpendicular to the vector `4 hat i + 3 hat j + 2 hat k`
Find the vector equation of the plane passing through the points `hati +hatj-2hatk, hati+2hatj+hatk,2hati-hatj+hatk`. Hence find the cartesian equation of the plane.
Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector `2hati + hatj + 2hatk.`
Parametric form of the equation of the plane is `bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk)` λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.
Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and z-axis respectively.
Find the vector equation of a line passing through the points A(3, 4, –7) and B(6, –1, 1).
Find the Cartesian equation of the following planes:
`vecr.(hati + hatj-hatk) = 2`
Find the Cartesian equation of the following planes:
`vecr.(2hati + 3hatj-4hatk) = 1`
Find the Cartesian equation of the following planes:
`vecr.[(s-2t)hati + (3 - t)hatj + (2s + t)hatk] = 15`
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
2x + 3y + 4z – 12 = 0
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
3y + 4z – 6 = 0
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
5y + 8 = 0
Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + 2 \hat{k} \right) = 6\]
Find the vector and Cartesian forms of the equation of the plane passing through the point (1, 2, −4) and parallel to the lines \[\vec{r} = \left( \hat{i} + 2 \hat{j} - 4 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right)\] and \[\vec{r} = \left( \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)\] Also, find the distance of the point (9, −8, −10) from the plane thus obtained.
Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + \hat{k} \right) = 6 .\]
Find the vector and cartesian equations of the plane passing throuh the points (2,5,- 3), (-2, - 3,5) and (5,3,-3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (–1, –3, –1).
Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector `2hati + 2hatj - 3hatk`.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The Cartesian equation of the plane `vec"r" * (hat"i" + hat"j" - hat"k")` = 2 is ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is `vec"r" = 5hat"i" - 4hat"j" + 6hat"k" + lambda(3hat"i" + 7hat"j" + 2hat"k")`.
Find the vector and the cartesian equations of the plane containing the point `hati + 2hatj - hatk` and parallel to the lines `vecr = (hati + 2hatj + 2hatk) + s(2hati - 3hatj + 2hatk)` and `vecr = (3hati + hatj - 2hatk) + t(hati - 3hatj + hatk)`
