Chapter 9.7, Problem 4AYU

True or False $u\times v$ is a vector that is parallel to both $u\text{ and }v$.

To determine

True or False, $\text{u} \times \text{v}$ is a vector, that is parallel to both $\text{u}$ and $\text{v}$.

Answer to Problem 4AYU

Solution:

False

Explanation of Solution

Given: $\text{u} \times \text{v}$ is a vector

Let $\text{u} = {\text{a}}_1\text{i} + {\text{b}}_1\text{j} + {\text{c}}_1\text{k}$ and $\text{v} = {\text{a}}_2\text{i} + {\text{b}}_2\text{j} + {\text{c}}_2\text{k}$, then

Now compute the dot product $\text{u}$. $\left(\text{u} \times \text{v}\right)$,

$u.\left(u\times v\right)=\left(a_{1}i+b_{1}j+c_{1}k\right).\left[\left(b_1{c}_2-b_2{c}_1\right)i - \left(a_1{c}_2-a_2{c}_1\right)j+\left(a_1{b}_2-b_2{a}_1\right)k\right]$

$=a_1\left(b_1{c}_2-b_2{c}_1\right)–b_1\left(a_1{c}_{2 }-a_2{c}_1\right)+c_1\left(a_1{b}_2-b_2{a}_1\right)$

$u.\left(u\times v\right)=0$

Since two vectors are orthogonal, if their dot product is zero.

it follows that $\text{u}$ and $\text{u} \times \text{v}$ are orthogonal.

Similarly, $\text{v}.\left(\text{u} \times \text{v}\right) = 0$

So $\text{v}$ and $\text{u} \times \text{v}$ are orthogonal.

$\text{u} \times \text{v}$ is orthogonal to both $$ \text{u}$$ and $$ \text{v}$$,

Hence, $\text{u} \times \text{v}$ is a vector that is not parallel to both $\text{u}$ and $\text{v}$.

Therefore, the answer is False.