JEE Exam > JEE Videos > Mathematics (Maths) Class 12 > Equation of a Plane passing through 3 Non-Collinear points and Equation of plane passing through the Intersection of 2 given Planes
Equation of a Plane passing through 3 Non-Collinear points and Equation of plane passing through the Intersection of 2 given Planes Video Lecture | Mathematics (Maths) Class 12 - JEE
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FAQs on Equation of a Plane passing through 3 Non-Collinear points and Equation of plane passing through the Intersection of 2 given Planes Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What is the equation of a plane passing through three non-collinear points? |  |
Ans. To find the equation of a plane passing through three non-collinear points, you can use the formula:
Ax + By + Cz + D = 0
where (x, y, z) are the coordinates of a point on the plane, and A, B, C, and D are constants. Substitute the coordinates of the three non-collinear points into this equation and solve the resulting system of equations to find the values of A, B, C, and D.
| 2. How do you determine if three points are non-collinear? | |
Ans. Three points are considered non-collinear if they do not lie on the same line. To determine if three points are non-collinear, you can calculate the area of the triangle formed by the points. If the area is non-zero, then the points are non-collinear. However, if the area is zero, the points are collinear.
| 3. What is the equation of a plane passing through the intersection of two given planes? | |
Ans. To find the equation of a plane passing through the intersection of two given planes, you can use the cross product of the normal vectors of the two planes. Let's say the normal vectors of the given planes are n1 and n2. Then the equation of the plane passing through their intersection is:
n1 · (x, y, z) = n1 · P
where (x, y, z) are the coordinates of a point on the plane, n1 · (x, y, z) represents the dot product between the normal vector and the coordinates, and P is a point on the intersection of the two planes.
| 4. How many points are needed to uniquely determine a plane? | |
Ans. Three non-collinear points are needed to uniquely determine a plane in three-dimensional space. These three points will not lie on the same line, ensuring that they define a unique plane. Any additional points beyond three will lie on the plane defined by these three points.
| 5. Can two planes intersect at a single point? | |
Ans. Yes, two planes can intersect at a single point. This occurs when the normal vectors of the two planes are not parallel. The intersection of the planes will be a line, and if this line intersects with a third plane, it can result in a single point of intersection. However, it is also possible for the two planes to be parallel or coincide, in which case they will not intersect or intersect along a line, respectively.
Text Transcript from Video
equation of a plane passing through
three non collinear points let are s
tb-3 non collinear points on the plane
with position vectors a B C respectively
let P be any point with the position
vector R in the same plane now the
vectors RS and RT are in the given plane
by the definition of the cross product
RS cross R T is a vector perpendicular
to the plane containing the points are s
and T therefore the equation of the
plane passing through our and
perpendicular to the vector R s cross R
T is our vector minus a vector dot RS
vector cross RT vector is equal to zero
but vector RS is equal to B vector minus
a vector and RT is equal to C vector
minus a vector therefore vector equation
of the plane passing through three non
collinear points a B C is our vector
minus a vector dot B vector minus a
vector cross C vector minus a vector is
equal to zero now Cartesian form of the
equation of the plane passing through
three non collinear points let x1 y1 z1
x2 y2 z2 and x3 y3 z3 be the coordinates
of the point R s and T respectively and
also let XYZ be the coordinates of the
point P on the plane with position
vector R
now vector RP is equal to X minus x1
into I cap plus y minus y1 into J cap
plus Z minus z1 into k cap similarly we
can get the values of the vectors RS and
RT substituting these values in the
vector form of the equation of the plane
and expressing it in the determinant
form we get the equation of the plane
passing through three non collinear
points as determinant of X minus X 1 y
minus y1 Z minus z1 x2 minus x1 y2 minus
y1 z2 minus z1 x3 minus x1 y3 minus y1 Z
3 minus Z 1 is equal to 0 is it
necessary that the three points should
be always non collinear what is the
equation of the plane passing through
three collinear points planes passing
through three collinear points resembles
the pages of your book where the line
containing the points are s and T are
members in the binding of the book
therefore the required equation of the
plane will be the equation of straight
line passing through these three
collinear points
three non collinear points let are s
tb-3 non collinear points on the plane
with position vectors a B C respectively
let P be any point with the position
vector R in the same plane now the
vectors RS and RT are in the given plane
by the definition of the cross product
RS cross R T is a vector perpendicular
to the plane containing the points are s
and T therefore the equation of the
plane passing through our and
perpendicular to the vector R s cross R
T is our vector minus a vector dot RS
vector cross RT vector is equal to zero
but vector RS is equal to B vector minus
a vector and RT is equal to C vector
minus a vector therefore vector equation
of the plane passing through three non
collinear points a B C is our vector
minus a vector dot B vector minus a
vector cross C vector minus a vector is
equal to zero now Cartesian form of the
equation of the plane passing through
three non collinear points let x1 y1 z1
x2 y2 z2 and x3 y3 z3 be the coordinates
of the point R s and T respectively and
also let XYZ be the coordinates of the
point P on the plane with position
vector R
now vector RP is equal to X minus x1
into I cap plus y minus y1 into J cap
plus Z minus z1 into k cap similarly we
can get the values of the vectors RS and
RT substituting these values in the
vector form of the equation of the plane
and expressing it in the determinant
form we get the equation of the plane
passing through three non collinear
points as determinant of X minus X 1 y
minus y1 Z minus z1 x2 minus x1 y2 minus
y1 z2 minus z1 x3 minus x1 y3 minus y1 Z
3 minus Z 1 is equal to 0 is it
necessary that the three points should
be always non collinear what is the
equation of the plane passing through
three collinear points planes passing
through three collinear points resembles
the pages of your book where the line
containing the points are s and T are
members in the binding of the book
therefore the required equation of the
plane will be the equation of straight
line passing through these three
collinear points
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Dec 22, 2024 Last updated |
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