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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.
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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 which is at a distance of 5 units from the origin and which is normal to the vector `2hati + hatj + 2hatk.`
Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane `vec r.(hati+hatj+hatk)=2`
Find the vector equation of the plane which contains the line of intersection of the planes `vecr (hati+2hatj+3hatk)-4=0` and `vec r (2hati+hatj-hatk)+5=0` which is perpendicular to the plane.`vecr(5hati+3hatj-6hatk)+8=0`
Find the vector equation of the plane passing through three points with position vectors ` hati+hatj-2hatk , 2hati-hatj+hatk and hati+2hatj+hatk` . Also find the coordinates of the point of intersection of this plane and the line `vecr=3hati-hatj-hatk lambda +(2hati-2hatj+hatk)`
Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and z-axis respectively.
Find the equation of the plane which contains the line of intersection of the planes
`vecr.(hati-2hatj+3hatk)-4=0" and"`
`vecr.(-2hati+hatj+hatk)+5=0`
and whose intercept on x-axis is equal to that of on y-axis.
Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is `2hati-3hatj+6hatk`
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`
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.
x + y + z = 1
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
5y + 8 = 0
Find the equation of the plane through the line of intersection of `vecr*(2hati-3hatj + 4hatk) = 1`and `vecr*(veci - hatj) + 4 =0`and perpendicular to the plane `vecr*(2hati - hatj + hatk) + 8 = 0`. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.
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 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 Cartesian equation of the plane, passing through the line of intersection of the planes `vecr. (2hati + 3hatj - 4hatk) + 5 = 0`and `vecr. (hati - 5hatj + 7hatk) + 2 = 0` intersecting the y-axis at (0, 3).
Find the vector and cartesian equation of the plane passing through the point (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`.
Find the vector equation of the plane that contains the lines `vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k")` and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
