Prove that vectors u, v and w are coplanar if and only if vectors u, v and w are linearly dependent.
normal to plane of u and v =
i j k
ux uy uz
vx vy vz
UV = i(uy vz - vy uz)+j(uz vx - ux vz)+k (ux vy - uy vx)
normal to plane of u and w
UW =i(uy wz -wy uz) +j(uz wx - ux wz) +k (ux wy -uy wx)
now if those vectors are in the same direction then UW = k UV
To prove that vectors u, v, and w are coplanar if and only if they are linearly dependent, we need to show two things:
1. If vectors u, v, and w are coplanar, then they are linearly dependent.
2. If vectors u, v, and w are linearly dependent, then they are coplanar.
Let's start with the first part:
1. If vectors u, v, and w are coplanar, then they are linearly dependent.
To prove this, we need to show that if u, v, and w are coplanar, there exist scalars a, b, and c, not all zero, such that au + bv + cw = 0.
To begin, let's assume that u, v, and w are coplanar. This means that they lie in the same plane. Consider a point P in this plane, and let A be its position vector. Since u, v, and w are coplanar, we can express any vector in this plane as a linear combination of u, v, and w.
Now, let's consider the vector OP, where O is the origin. We can write OP as a linear combination of vectors u, v, and w, using the scalars a, b, and c:
OP = a*u + b*v + c*w
We want to show that there exist scalars a, b, and c, not all zero, such that OP = 0. If OP = 0, then a*u + b*v + c*w = 0.
If a, b, and c are all zero, then the vectors u, v, and w would be linearly independent. However, since we assumed that the vectors are coplanar, this is not possible.
Therefore, there must exist scalars a, b, and c, not all zero, such that a*u + b*v + c*w = 0. Hence, u, v, and w are linearly dependent.
2. If vectors u, v, and w are linearly dependent, then they are coplanar.
To prove this, we need to show that if u, v, and w are linearly dependent, then they lie in the same plane.
If u, v, and w are linearly dependent, there exist scalars a, b, and c, not all zero, such that a*u + b*v + c*w = 0.
Without loss of generality, assume that a ≠ 0. We can rewrite the equation as:
u = (-b/a)*v + (-c/a)*w
This equation shows that u is a linear combination of vectors v and w. Therefore, any vector that can be expressed as a linear combination of u, v, and w lies in the same plane as v and w.
Hence, vectors u, v, and w are coplanar.
In conclusion, we have shown that vectors u, v, and w are coplanar if and only if they are linearly dependent.