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Q.The non-parametric vector equation of a plane passing through a point whose position vector is a⃗\vec a and parallel to u⃗\vec u and v⃗\vec v, is :

(a) [r⃗−a⃗, u⃗, v⃗]=0[\vec r - \vec a,\ \vec u,\ \vec v] = 0
(b) [r⃗, u⃗, v⃗]=0[\vec r,\ \vec u,\ \vec v] = 0
(c) [r⃗, a⃗, u⃗×v⃗]=0[\vec r,\ \vec a,\ \vec u \times \vec v] = 0
(d) [a⃗, u⃗, v⃗]=0[\vec a,\ \vec u,\ \vec v] = 0
Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2017MCQ· 1mImportance★★★★★
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A general point r⃗\vec r lies in the plane through a⃗\vec a parallel to u⃗,v⃗\vec u,\vec v exactly when r⃗−a⃗\vec r-\vec a is a linear combination of u⃗\vec u and v⃗\vec v, i.e. when r⃗−a⃗,u⃗,v⃗\vec r-\vec a,\vec u,\vec v are coplanar — captured by a zero scalar triple product.

  1. Let the plane pass through the point AA with position vector a⃗\vec a, and be parallel to two non-parallel vectors u⃗\vec u and v⃗\vec v.
  2. Let r⃗\vec r be the position vector of an arbitrary point PP on the plane.
  3. The vector AP⃗=r⃗−a⃗\vec{AP} = \vec r - \vec a lies in the plane, hence it must be expressible as a linear combination of the two direction vectors: r⃗−a⃗=su⃗+tv⃗\vec r - \vec a = s\vec u + t\vec v for scalars s,ts,t (this is the parametric form). …
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