In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.

Verified step by step guidance

1. Understand that since vectors \( \mathbf{u} \) and \( \mathbf{v} \) have the same direction, \( \mathbf{v} \) is a scalar multiple of \( \mathbf{u} \). This means \( \mathbf{v} = k \mathbf{u} \) for some scalar \( k \).
2. Recall that the magnitude (or norm) of a vector \( \mathbf{v} \) is denoted by \( ||\mathbf{v}|| \) and is related to the magnitude of \( \mathbf{u} \) by \( ||\mathbf{v}|| = |k| \cdot ||\mathbf{u}|| \).
3. Identify or find the scalar \( k \) by comparing the components of \( \mathbf{v} \) and \( \mathbf{u} \), or by using any given information about the vectors in the problem.
4. Calculate the magnitude of \( \mathbf{u} \) using the formula \( ||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2 + u_3^2} \) if \( \mathbf{u} \) is in three dimensions, or the corresponding formula for the dimension given.
5. Multiply the magnitude of \( \mathbf{u} \) by the absolute value of \( k \) to find \( ||\mathbf{v}|| = |k| \cdot ||\mathbf{u}|| \).

Key Concepts

Vector Direction and Scalar Multiplication

When two vectors have the same direction, one vector can be expressed as a scalar multiple of the other. This means v = k * u, where k is a scalar. Understanding this relationship helps in determining the magnitude of one vector based on the other.

Magnitude (Norm) of a Vector

The magnitude or norm of a vector, denoted ||v||, represents its length in space. It is calculated using the square root of the sum of the squares of its components. Knowing how to compute or relate magnitudes is essential for solving problems involving vector lengths.

Properties of Vectors in the Same Direction

Vectors pointing in the same direction have proportional components and their magnitudes relate by the absolute value of the scalar multiple. This property allows one to find the magnitude of one vector if the other vector and the scalar factor are known.