Show that if and only if A.B=0
To prove the statement "if and only if A.B = 0," we need to show two things:
1. If A.B = 0, then A and B are orthogonal.
2. If A and B are orthogonal, then A.B = 0.
Let's start with statement 1.
1. If A.B = 0, then A and B are orthogonal:
Orthogonal vectors are defined as vectors that have a dot product of zero.
So, if A.B = 0, it means that the dot product of A and B is zero, which implies that A and B are orthogonal.
Statement 1 is proven.
Now, let's move on to statement 2.
2. If A and B are orthogonal, then A.B = 0:
Suppose A and B are orthogonal. By definition, this means that their dot product is zero.
We can express the dot product A.B as:
A.B = |A| |B| cos(theta)
where |A| and |B| are the magnitudes of vectors A and B, respectively, and theta is the angle between them.
Since A and B are orthogonal, cos(theta) = 0, which means that the dot product A.B is equal to zero.
Statement 2 is proven.
Therefore, we have shown both directions of the "if and only if" statement, and we can conclude that A.B = 0 if and only if A and B are orthogonal.
To prove that A.B = 0 if and only if A and B are orthogonal, we need to show two things:
1. If A and B are orthogonal, then A.B = 0.
2. If A.B = 0, then A and B are orthogonal.
Let's prove each of these statements step by step:
1. If A and B are orthogonal, then A.B = 0:
By definition, two vectors A and B are orthogonal if their dot product A.B is equal to zero.
Let's assume that A and B are orthogonal. Then, A.B = 0.
Therefore, if A and B are orthogonal, then A.B = 0.
2. If A.B = 0, then A and B are orthogonal:
Let's assume that A.B = 0.
To prove that A and B are orthogonal, we need to show that the angle between them is 90 degrees or π/2 radians.
We can use the definition of the dot product to determine the angle between A and B:
A.B = |A| |B| cos(theta),
where |A| and |B| are the magnitudes of vectors A and B, and theta is the angle between them.
Since A.B = 0, the equation becomes:
0 = |A| |B| cos(theta).
If cos(theta) = 0, then theta = π/2 or 90 degrees.
Therefore, A and B are orthogonal if and only if A.B = 0.
By proving both statements, we have shown that if and only if A.B = 0, A and B are orthogonal.