Chapter 9, Problem 110RE

To determine

To calculate: The cross product of vectors $u\times v$ where vector $u=3v$ .

Answer to Problem 110RE

The cross product of vectors $v\times u=\left(0i+0j+0k\right)$ .

Explanation of Solution

Given information:

The vectors are given as:

  $u=3v$

Formula used:

let $u$ and $v$ are two vectors and $\alpha$ is scalar,

then, algebraic property of the cross product is $\alpha \left(u\times v\right)=\alpha u\times v$

The cross product of vectors $u=a_{1}i+b_{1}j+c_{1}k$ and $v=a_{2}i+b_{2}j+c_{2}k$ is,

  $\begin{array}{c}u\times v=\left|\begin{array}{ccc}i& j& k\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right|\\ =i\left|\begin{array}{cc}{b}_1& c_1\\ b_2& c_2\end{array}\right|-j\left|\begin{array}{cc}{a}_1& c_1\\ a_2& c_2\end{array}\right|+k\left|\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right|\\ =i\left(b_1{c}_2-c_1{b}_2\right)-j\left(a_1{c}_2-c_1{a}_2\right)+k\left(a_1{b}_2-b_1{a}_2\right)\end{array}$

Calculation:

Consider the provided vectors,

  $u=3v$

Apply the algebraic property of the cross product,

  $\begin{array}{c}3\left(u\times v\right)=3u\times v\\ =v\times v\end{array}$

Let consider the vector $v=a_{1}i+b_{1}j+c_{1}k$ and apply the algebraic cross product rule,

  $\begin{array}{c}v\times v=\left|\begin{array}{ccc}i& j& k\\ a_1& b_1& c_1\\ a_1& b_1& c_1\end{array}\right|\\ =i\left|\begin{array}{cc}{b}_1& c_1\\ b_1& c_1\end{array}\right|-j\left|\begin{array}{cc}{a}_1& c_1\\ a_1& c_1\end{array}\right|+k\left|\begin{array}{cc}{a}_1& b_1\\ a_1& b_1\end{array}\right|\\ =i\left(b_1{c}_1-c_1{b}_1\right)-j\left(a_1{c}_1-c_1{a}_1\right)+k\left(a_1{b}_1-b_1{a}_1\right)\\ =0i+0j+0k\end{array}$

Thus, the cross product of vectors is $v\times u=\left(0i+0j+0k\right)$