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The magnitude of a vector, sometimes referred to as the length or norm of the vector, is a measure of its size. If you imagine a vector as an arrow pointing from one point to another in space, the magnitude is simply the length of that arrow. We denote the magnitude of a vector \( \mathbf{a} \) as \( \| \mathbf{a} \| \).
To calculate the magnitude of a vector, you use the formula:

• For 2-dimensional vectors: \( \| \mathbf{a} \| = \sqrt{x^2 + y^2} \)
• For 3-dimensional vectors: \( \| \mathbf{a} \| = \sqrt{x^2 + y^2 + z^2} \)

The magnitude is always a non-negative number, meaning it is either zero or a positive value. If the magnitude of a vector is zero, it means the vector is essentially a point rather than an arrow, signifying no movement away from the origin. This understanding is crucial in analyzing vector relationships, like proving when two vectors are equal based on their distance.

Vector subtraction is an essential operation when working with vectors. It is used to find the difference between two vectors. To subtract one vector \( \mathbf{v} \) from another vector \( \mathbf{u} \), you subtract each corresponding component of \( \mathbf{u} \) and \( \mathbf{v} \). Formally, if \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \), then:

• \( \mathbf{u} - \mathbf{v} = (u_1 - v_1, u_2 - v_2, u_3 - v_3) \)

The result of this subtraction is a new vector that represents the direct "distance" and direction from \( \mathbf{v} \) to \( \mathbf{u} \). If you set \( \mathbf{u} - \mathbf{v} = \mathbf{0} \) (the zero vector), it indicates \( \mathbf{u} \) and \( \mathbf{v} \) are identical, thus proving they are the same point in space.

Proof by implication is a logical process used to demonstrate that a certain statement is true if another condition holds. It involves showing that one statement leads to or results in another. In mathematics, this is expressed as "If \( A \), then \( B \)", and is often shown using real-world or mathematical examples.
Let's apply this to our exercise: we have two directions to prove:

• First, if \( \mathrm{d}(\mathbf{u}, \mathbf{v}) = 0 \), then it implies \( \mathbf{u} = \mathbf{v} \).
• Second, if \( \mathbf{u} = \mathbf{v} \), then \( \mathrm{d}(\mathbf{u}, \mathbf{v}) = 0 \).

In both cases, you show that each condition naturally leads to the opposite. This kind of reasoning confirms the 'if and only if' nature of the statement, ensuring it's valid in both directions. Understanding proof by implication helps in unraveling the logical relationships within mathematical statements and is a key method in rigorous mathematical argumentation.