Chapter 10, Problem 7RWDT

To determine

To prove: Every nonzero vector can be written as a positive constant time in a unit vector.

Answer to Problem 7RWDT

It is proved that every nonzero vector can be written as a positive constant time a unit vector, and the positive constant is precisely the length of the vector.

Explanation of Solution

Given information: Consider a nonzero vector.

Calculation:

Let $u$ be a nonzero vector then $\frac{u}{\left|u\right|}$ is a unit vector (vector of length 1), where $\left|u\right|$ is the length of the vector. The unit vector $\frac{u}{\left|u\right|}$ also represents the direction of the vector $u$ . So,

  $u=\left|u\right|\left(\frac{u}{\left|u\right|}\right)$

Therefore, $u$ can be written as a positive constant $\left(\left|u\right|>0\right)$ times the unit vector. Since the length of the unit of unit vector $1$ . In the above equation, the length of the vectors on both sides must be the same. Thus, the positive constant must be $\left|u\right|$ .

Therefore, every nonzero vector can be written as a positive constant time a unit vector, and the positive constant is precisely the length of the vector.