Proof: Vectors 𝑢, 𝑣 , and 𝑤 are coplanar, only if vectors 𝑢, 𝑣 , and 𝑤 are linearly
dependent:
In order for 𝑢, 𝑣 , and 𝑤 to be coplanar, they can be written as linear combination, so that
𝑢 = 𝑠𝑣 + 𝑡𝑤 (for two non zero scalars s, and t). For this first proof, we assume that
vectors 𝑣 , and 𝑤 are not collinear:
So therefore:
𝑢 = 𝑠𝑣 + 𝑡𝑤 … … . . (1)
Slightly rearranging equation (1), we get:
𝑢 − 𝑠𝑣 − 𝑡𝑤 = 𝑠𝑣 − 𝑠𝑣 + 𝑡𝑤 − 𝑡𝑤
𝑢 − 𝑠𝑣 − 𝑡𝑤 = 0 … . (2)
Since the coefficient for the vector 𝑢 is 1, a non zero constant, we know that vectors
𝑢, 𝑣 , and 𝑤 are linearly dependent.
If the two vectors, 𝑣 , and 𝑤 are collinear, then
𝑤 = 𝑠𝑣 … . (3)
Proof: If two collinear vectors are coplanar, they must also be linearly dependent: To
prove that these two vectors are linearly dependent, we mu
Slightly rearranging equation (3), we get:
𝑤 − 𝑠𝑣 = 𝑠𝑣 − 𝑠𝑣
1𝑤 − 𝑠𝑣 = 0
(multiply each side by 0 𝑢)
1𝑤 − 𝑠𝑣 − 0𝑢 = 0
Since the vector, 𝑤 has 1 as a coefficient, a non zero constant, we know that that when
the two vectors 𝑣 , and 𝑤 are collinear, the vectors 𝑢, 𝑣 , and 𝑤 are linearly dependent.
Proof: The three linearly dependent vectors, 𝑢, 𝑣 , and 𝑤 are also coplanar:
𝑢, 𝑣 , and 𝑤 are linearly dependent:, we use equation (2):
𝑢 − 𝑠𝑣 − 𝑡𝑤 = 0
Slightly rearranging the equation above, we get:
𝑢 − 𝑠𝑣 + 𝑠𝑣 − 𝑡𝑤 + 𝑡𝑤 = 0 + 𝑠𝑣 + 𝑡𝑤
𝑢 = 𝑠𝑣 + 𝑡𝑤 (The same equation as (1): Please note: the scalars s, and t are non zero)
As you can see, the vectors 𝑣 , and 𝑤, can be written as a linear combination of vector 𝑢,
illustrating that the linearly dependent vectors 𝑢, 𝑣 , and 𝑤 are coplanar.