In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

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Understand that two vectors \( \mathbf{u} \) and \( \mathbf{v} \) having the same direction means they are scalar multiples of each other. This implies \( \mathbf{u} = k \mathbf{v} \) for some scalar \( k > 0 \).

Recall that for \( \mathbf{u} = \mathbf{v} \) to be true, both the magnitude (length) and direction of \( \mathbf{u} \) and \( \mathbf{v} \) must be exactly the same.

Since \( \mathbf{u} \) and \( \mathbf{v} \) have the same direction, check if their magnitudes are equal by comparing \( |\mathbf{u}| \) and \( |\mathbf{v}| \).

If \( |\mathbf{u}| = |\mathbf{v}| \), then \( k = 1 \) and \( \mathbf{u} = \mathbf{v} \). Otherwise, if \( |\mathbf{u}| \neq |\mathbf{v}| \), then \( \mathbf{u} \neq \mathbf{v} \) even though they point in the same direction.

Summarize your conclusion by stating that having the same direction does not guarantee \( \mathbf{u} = \mathbf{v} \); equality requires both direction and magnitude to be identical.

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