Question

Find the vector equation of a plane

a) Passing through the point A(-1, 2, -3) and parallel to the vectors u ⃗=(-2,1,0) and
v ⃗=(2,-3,-1).
b) Passing through the points A(2, 3, 2) and B(2, 1, 5) and C(3, -1, 0)
c) Passing through the origin and containing the line r ⃗=(1,-3,2)+s(1,1,1).

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Answer 1

To find the vector equations of planes in different scenarios, normal vectors are obtained through cross products of direction vectors or points on the plane, which is then used along with a point on the plane to write the equation.

Step-by-step explanation:

To find the vector equation of a plane that satisfies the given conditions, we can use the following approaches:

1. For a plane passing through the point A(-1, 2, -3) and parallel to vectors u ⃗=(-2,1,0) and v ⃗=(2,-3,-1), we need to first find a normal vector to the plane by calculating the cross product of u ⃗ and v ⃗. The cross product will give us the components of the normal vector, which we can use to write the equation of the plane.
2. For a plane passing through points A(2, 3, 2), B(2, 1, 5), and C(3, -1, 0), we need to calculate two direction vectors using the positions of the points, and then find the cross product to get the normal vector. Once we have the normal vector, we can form the equation of the plane.
3. For a plane passing through the origin and containing the line r ⃗=(1,-3,2)+s(1,1,1), we observe that one direction vector is given directly by the line's directional vector, and since the plane passes through the origin, any position vector on the line can be used as a second direction vector. Using this information, we can find the normal vector through the cross product and then write the equation of the plane.