The process to determine if three given vectors are co-planar involves evaluating their scalar triple product. The scalar triple product of three vectors (a, b, c) is reached by taking the dot product of one of the vectors with the cross product of the remaining two. Co-planarity of the vectors is established if the scalar triple product is equal to zero.
Here are the vectors:
v1 = (7, 8, 9)
v2 = (4, 0, 3)
v3 = (3, 8, 6)
For vectors v1, v2, and v3, we first need to compute the cross product of vectors v2 and v3. The cross product of two vectors (u1, u2, u3) and (v1, v2, v3) is given by:
(u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)
Substituting values, the cross product of vectors v2 and v3 is (0*6 - 3*8, 3*3 - 4*6, 4*8 - 0*3) = (-24, -9, 32),
Next, we compute the dot product of v1 and this cross product. The dot product of two vectors (u1, u2, u3) and (v1, v2, v3) is given by:
u1v1 + u2v2 + u3v3
Substituting the values, the dot product of v1 and the cross product of v2 and v3 is 7*(-24) + 8*(-9) + 9*32 = -168 - 72 + 288 = 48
Since the scalar triple product is NOT zero (it's 48), the vectors are NOT co-planar. Hence the statement "the vectors (7,8,9), (4,0,3), (3,8,6) are coplanar" is False.