Condition for Two Non-Zero Vectors to be Parallel

Two non-zero vectors A and B are said to be parallel if they have the same direction or are in the same line. In other words, vectors A and B are parallel if one is a scalar multiple of the other. Mathematically, the condition for vectors A and B to be parallel can be expressed as:

A = k * B

where k is a scalar constant.

Condition for Two Non-Zero Vectors to be Orthogonal (Perpendicular)

Two non-zero vectors A and B are said to be orthogonal or perpendicular if their dot product is zero. The dot product of two vectors is the sum of the products of their corresponding components. Mathematically, the condition for vectors A and B to be orthogonal can be expressed as:

A · B = 0

where "·" denotes the dot product.

Combining the Conditions

To determine the condition for two non-zero vectors A and B to be both parallel and orthogonal, we need to satisfy both the conditions mentioned above simultaneously.

Let's assume that vectors A and B are parallel and orthogonal. Using the condition for parallel vectors, we have:

A = k * B

Taking the dot product of both sides with vector B, we get:

A · B = (k * B) · B

Using the properties of the dot product, we can simplify the equation:

A · B = k * (B · B)

Since A and B are orthogonal, A · B = 0. Substituting this in the equation, we have:

0 = k * (B · B)

For B · B ≠ 0, the only way this equation can hold is if k = 0. However, if k = 0, then A will be the zero vector, which contradicts the given condition that A and B are non-zero vectors.

Hence, the condition for two non-zero vectors A and B to be both parallel and orthogonal is that they must be linearly dependent, i.e., one vector must be a scalar multiple of the other.