Equation of a plane passing through three non-collinear points is Note Where (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3) are coordinates of three non-collinear points.

Question

Equation of a plane passing through three non-collinear points is Note Where (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3) are coordinates of three non-collinear points. (A) ( r - a ) ⋅[( b - a ) ×( c - a )]=0, where a , b , c are the position vectors of three non-collinear points (B) |x-x1 y-y1 z-z1 x2-x1 y2-y1 z2-z1 x3-x1 y3-y1 z3-z1|=0 (C) Both(a) and (b) (D) None of the above

Tardigrade Admin · Accepted Answer

Let R, S and T be three non-collinear points on the plane with position vectors a, b and c, respectively.

The vector RS and RT are in the given plane. Therefore, the vector RS × RT is perpendicular to the plane containing points R, S and T. Let r be the position vector of any point P in the plane. Therefore, the equation of the plane passing through R and perpendicular to the vector RS × RT is (r-a) ⋅(R S × R T) =0 text (if a perp b text , then a ⋅ b=0) ⇒ (r-a) ⋅[(b-a) ×(c-a)] =0 ...(i) This is the equation of the plane in vector form passing through three non-collinear points. Note Why was it necessary to say that the three points had to be non-collinear? If the three points were on the same line, then there will be many planes that will contain them.

These planes will resemble the pages of a book where the line containing the points R, S and T are members in the binding of the book. Cartesian Form Let (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3) be the coordinates of the points R, S and T, respectively. Let (x, y, z) be the coordinates of any point P on the plane with position vector r. Then, RP =(x-x1) hat i +(y-y1) hat j +(z-z1) hat k RS =(x2-x1) hat i +(y2-y1) hat j +(z2-z1) hat k RT =(x3-x1) hat i +(y3-y1) hat j +(z3-z1) hat k Substituting these values in Eq. (i) of the vector form and expressing it in the form of a determinant, we have |x-x1 y-y1 z-z1 x2-x1 y2-y1 z2-z1 x3-x1 y3-y1 z3-z1|=0 which is the equation of the plane in cartesian form passing through three non-collinear points (x1, y1, z1),(x2, y2, z2) and (x3, y3, z3)

Q. Equation of a plane passing through three non-collinear points is
Note Where $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$ and $(x_{3}, y_{3}, z_{3})$ are coordinates of three non-collinear points.

$(r - a) \cdot [(b - a) \times (c - a)] = 0$ , where $a, b, c$ are the position vectors of three non-collinear points

$∣ ∣ x - x_{1} x_{2} - x_{1} x_{3} - x_{1} y - y_{1} y_{2} - y_{1} y_{3} - y_{1} z - z_{1} z_{2} - z_{1} z_{3} - z_{1} ∣ ∣ = 0$

Both(a) and (b)

None of the above

Q. Equation of a plane passing through three non-collinear points is Note Where (x1​,y1​,z1​),(x2​,y2​,z2​) and (x3​,y3​,z3​) are coordinates of three non-collinear points.

(r−a)⋅[(b−a)×(c−a)]=0, where a,b,c are the position vectors of three non-collinear points

∣∣​x−x1​x2​−x1​x3​−x1​​y−y1​y2​−y1​y3​−y1​​z−z1​z2​−z1​z3​−z1​​∣∣​=0

Both(a) and (b)

None of the above

Q. Equation of a plane passing through three non-collinear points is
Note Where $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$ and $(x_{3}, y_{3}, z_{3})$ are coordinates of three non-collinear points.

$(r - a) \cdot [(b - a) \times (c - a)] = 0$ , where $a, b, c$ are the position vectors of three non-collinear points

$∣ ∣ x - x_{1} x_{2} - x_{1} x_{3} - x_{1} y - y_{1} y_{2} - y_{1} y_{3} - y_{1} z - z_{1} z_{2} - z_{1} z_{3} - z_{1} ∣ ∣ = 0$