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

Verified step by step guidance

1. Understand that two vectors \( \mathbf{u} \) and \( \mathbf{v} \) having the same direction means they are scalar multiples of each other, i.e., \( \mathbf{u} = k \mathbf{v} \) for some positive scalar \( k \).
2. Recall that for \( \mathbf{u} = \mathbf{v} \) to be true, both the magnitude and direction of \( \mathbf{u} \) and \( \mathbf{v} \) must be exactly the same.
3. Since the problem states \( \mathbf{u} \) and \( \mathbf{v} \) have the same direction, check if their magnitudes are equal by comparing \( \| \mathbf{u} \| \) and \( \| \mathbf{v} \| \).
4. If \( \| \mathbf{u} \| = \| \mathbf{v} \| \), then \( \mathbf{u} = \mathbf{v} \) because they have the same direction and magnitude.
5. If \( \| \mathbf{u} \| \neq \| \mathbf{v} \| \), then \( \mathbf{u} \neq \mathbf{v} \) even though they point in the same direction, because their lengths differ.

Key Concepts

Vector Direction

Vector direction refers to the orientation of a vector in space, independent of its magnitude. Two vectors have the same direction if they lie along the same line or parallel lines, pointing either the same way or exactly opposite. Understanding direction is crucial to compare vectors beyond just their lengths.

Vector Equality

Two vectors are equal if and only if they have the same magnitude and the same direction. Even if vectors share the same direction, they are not equal unless their lengths are identical. This concept helps determine whether u equals v when their directions match.

Scalar Multiplication of Vectors

Scalar multiplication changes a vector's magnitude without altering its direction, unless the scalar is negative, which reverses the direction. Recognizing that vectors with the same direction can differ by a scalar factor is essential to analyze if u equals v or if one is a scaled version of the other.