Show that if and only if A.B=0?

Question

2. Let A1, A2, …, Ar be non-zero vectors such that Ai.Aj = o if i is different from j.
Let c1, c2, …., cr be numbers such that c1A1+ c2A2+…..+ crAr = 0. Show that ci = 0 for all i = 1,2, 3, …, r.

Answer 1

To prove that A.B = 0 if and only if A and B are orthogonal, we will show both directions of the statement.

Direction 1: If A.B = 0, then A and B are orthogonal.
Assume A.B = 0.
This means that the dot product of A and B is zero.
By the definition of dot product, we have A.B = ||A|| ||B|| cos(theta), where ||A|| and ||B|| are the magnitudes (or lengths) of A and B, and theta is the angle between A and B.
Since the dot product A.B is zero, we have ||A|| ||B|| cos(theta) = 0.
Since ||A|| and ||B|| are non-zero (given that A and B are non-zero vectors), we can conclude that cos(theta) must be zero.
Cos(theta) is equal to zero when theta is equal to 90 degrees or pi/2 radians.
Therefore, A and B are orthogonal.

Direction 2: If A and B are orthogonal, then A.B = 0.
Assume A and B are orthogonal.
This means that the angle between A and B is 90 degrees or pi/2 radians.
Using the definition of dot product, we have A.B = ||A|| ||B|| cos(theta), where ||A|| and ||B|| are the magnitudes (or lengths) of A and B, and theta is the angle between A and B.
Since cos(theta) is zero for a 90-degree angle, we have cos(theta) = 0.
Therefore, A.B = ||A|| ||B|| cos(theta) = ||A|| ||B|| (0) = 0.

Therefore, we have shown that A.B = 0 if and only if A and B are orthogonal.

Now, let's prove that ci = 0 for all i = 1,2, 3, ..., r when c1A1 + c2A2 + ... + crAr = 0.

Assume c1A1 + c2A2 + ... + crAr = 0.
Since Ai.Aj = 0 for i ≠ j, we can rewrite the above equation as:
c1A1.A1 + c2A2.A2 + ... + crAr.Ar = 0
c1 ||A1||^2 + c2 ||A2||^2 + ... + cr ||Ar||^2 = 0 (using the definition of dot product)
Since Ai ≠ 0 (given that Ai is non-zero), we can conclude that ||Ai||^2 ≠ 0 for all i = 1,2, 3, ..., r.
Therefore, in order for the above equation to hold true (c1 ||A1||^2 + c2 ||A2||^2 + ... + cr ||Ar||^2 = 0), all ci's must be zero.
Hence, ci = 0 for all i = 1,2, 3, ..., r when c1A1 + c2A2 + ... + crAr = 0.

Therefore, we have proven that ci = 0 for all i = 1,2, 3, ..., r when c1A1 + c2A2 + ... + crAr = 0.