Final answer:
Basis for the plane P represented by 7x-8y+9z = 0, one can choose two linearly independent vectors lying within that plane, like (1, 7/8, 0) and (8/7, 1, 0).
Step-by-step explanation:
The question requires finding a basis for the plane P represented by the equation 7x−8y+9z = 0. A basis for a plane in three-dimensional space is a set of two linearly independent vectors that lie within the plane. To find a basis, we can begin by setting two of the variables to zero in turn, which will give us the vectors along the plane when the third variable is non-zero.
For example, if we set y = 0 and z = 0, we can solve for x. The equation becomes 7x = 0, which gives us x = 0. However, this does not yield a useful basis vector since it results in the zero vector (0, 0, 0). Instead, let's set z = 0 and find a vector for when x and y are non-zero:
- Let z = 0 and x = 1, which gives 7(1) − 8y + 9(0) = 0, so y = 7/8. Therefore, one basis vector is (1, 7/8, 0).
- Next, let z = 0 and y = 1, which gives 7x - 8(1) + 9(0) = 0, so x = 8/7. This produces another basis vector (8/7, 1, 0).
So a possible basis for the plane P could be the vectors (1, 7/8, 0) and (8/7, 1, 0). It is important to note that there are infinitely many choices for a basis for a plane, but any other basis will consist of two linearly independent vectors that can be written as linear combinations of the above two vectors.