Chapter 11.3, Problem 67E

To determine

Whether the statement “If the magnitude of two vectors are doubled, then the magnitude of their cross product is also doubled.” is true or false.

Answer to Problem 67E

The statement “If the magnitude of two vectors are doubled, then the magnitude of their cross product is also doubled.” is false.

Explanation of Solution

Given information:

The statement “If the magnitude of two vectors are doubled, then the magnitude of their cross product is also doubled.”

Consider the provided statement “If the magnitude of two vectors are doubled, then the magnitude of their cross product is also doubled.”

Let $a$ and $b$ be any two arbitrary vectors, then the cross product of the vectors is denoted by $a\times b$ .

Mathematically the magnitude of cross product is expressed as, $\Vert a\times b\Vert =\Vert a\Vert \cdot \Vert b\Vert \cdot \sin \theta$ , where $\Vert a\Vert$ and $\Vert b\Vert$ denote the magnitude of vectors $a$ and $b$ respectively and $\theta$ is the angle between them.

According to the question, magnitude of vectors are doubled, therefore, $\Vert a\Vert =\Vert 2a\Vert$ and $\Vert b\Vert =\Vert 2b\Vert$ .

Now, the magnitude of cross product is,

  $\begin{array}{c}\Vert a\times b\Vert =\Vert 2a\Vert \cdot \Vert 2b\Vert \cdot \sin \theta \\ =2\Vert a\Vert \cdot 2\Vert b\Vert \cdot \sin \theta \\ =4\Vert a\Vert \cdot \Vert b\Vert \cdot \sin \theta \end{array}$

If the magnitude of two vectors are doubled, then the magnitude of their cross product becomes four time of the original one.

Thus, the statement “If the magnitude of two vectors are doubled, then the magnitude of their cross product is also doubled.” is false.