Chapter 11.3, Problem 65E

To determine

Whether the statement “The cross product is not defined for vectors in the plane.” is true or false.

Answer to Problem 65E

The statement “The cross product is not defined for vectors in the plane.” is true.

Explanation of Solution

Given information:

The statement “The cross product is not defined for vectors in the plane.”

Consider the provided statement “The cross product is not defined for vectors in the plane.”

According to the definition of cross product if there are two three-dimensional vectors in a space given in the form of their components, say, $a=\langle a_1,a_2,a_3\rangle$ and $b=\langle b_1,b_2,b_3\rangle$ , then the cross product of two vectors is defined as ,

  $\begin{array}{c}a\times b=\begin{array}{ccc}i& j& k\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\\ \text{= }\left(a_2{b}_3-a_3{b}_2\right)i-\left(a_1{b}_3-a_3{b}_1\right)j+\left(a_1{b}_2-a_2{b}_1\right)k\end{array}$

Therefore, the cross product is not defined for vectors in two dimensional space.

Thus, the statement “The cross product is not defined for vectors in the plane.” is true.