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

Verified step by step guidance

1. insert step 1: Understand that vectors u and v having the same direction means they are scalar multiples of each other.
2. insert step 2: Express vector u as u = k * v, where k is a scalar.
3. insert step 3: Consider the condition for u = v, which is when the scalar k equals 1.
4. insert step 4: Check if the magnitudes of u and v are equal, as this would imply k = 1.
5. insert step 5: Conclude that if the magnitudes are equal, then u = v; otherwise, u is not equal to v.

Key Concepts

Vector Direction

In trigonometry and vector analysis, the direction of a vector is defined by the angle it makes with a reference axis. Two vectors are said to have the same direction if they point in the same way, regardless of their magnitudes. This concept is crucial for understanding vector equality, as direction plays a key role in determining whether two vectors can be considered equivalent.

Vector Equality

Two vectors are equal if they have the same magnitude and direction. This means that even if two vectors point in the same direction, they are only equal if their lengths are also identical. Understanding this concept is essential for answering the question, as it directly addresses whether u and v can be considered equal based solely on their directional alignment.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem in a Cartesian coordinate system. It is represented as a non-negative value and is independent of the vector's direction. In the context of the question, knowing the magnitudes of u and v is necessary to determine if they are equal, as having the same direction alone is insufficient for equality.