Chapter 9.5, Problem 2E

To determine

To describe: The cross product theorem and the properties of cross product.

Answer to Problem 2E

The cross product of the vectors u and v is orthogonal to the vector u and v. The vectors u and v lie in a plane then the vector $u\times v$ is orthogonal to the plane.

Explanation of Solution

Given:

The two vectors are $u=\langle a_1,a_2,a_3\rangle$ and $v=\langle b_1,b_2,b_3\rangle$.

Formula used: The cross product of the vectors $u=\langle a_1,a_2,a_3\rangle$ and $v=\langle b_1,b_2,b_3\rangle$ is given below,

$u\times v=\left(a_2{b}_3-a_3{b}_2\right)i-\left(a_1{b}_3-a_3{b}_1\right)j+\left(a_1{b}_2-a_2{b}_1\right)k$ (1)

The Dot Product of two vectors $u=\langle a_1,a_2,a_3\rangle$ and $v=\langle b_1,b_2,b_3\rangle$ is given below

$u\cdot v=a_1{b}_1+a_2{b}_2+a_3{b}_3$ (2)

Calculation:

The two vectors are orthogonal if the dot product of the two vectors is zero.

From equation (1) and (2), the dot product of $u\times v$ and u is given below,

$\begin{array}{c}\left(u\times v\right)\cdot u=\left(a_2{b}_3-a_3{b}_2\right)a_1-\left(a_1{b}_3-a_3{b}_1\right)a_2+\left(a_1{b}_2-a_2{b}_1\right)a_3\\ =a_1{a}_2{b}_3-a_1{a}_3{b}_2-a_1{a}_2{b}_3+a_2{a}_3{b}_1+a_1{a}_3{b}_2-a_2{a}_3{b}_1\\ =0\end{array}$

The dot product of $u\times v$ and u is 0, therefore $u\times v$ and u is orthogonal to each other.

From equation (1) and (2), the dot product of $u\times v$ and v is given below,

$\begin{array}{c}\left(u\times v\right)\cdot v=\left(a_2{b}_3-a_3{b}_2\right)b_1-\left(a_1{b}_3-a_3{b}_1\right)b_2+\left(a_1{b}_2-a_2{b}_1\right)b_3\\ =a_2{b}_1{b}_3-a_3{b}_1{b}_2-a_1{b}_2{b}_3+a_3{b}_2{b}_1+a_1{b}_2{b}_3-a_2{b}_1{b}_3\\ =0\end{array}$

The dot product of $u\times v$ and v is 0, therefore $u\times v$ and v is orthogonal to each other.

The vector $u\times v$ is orthogonal to the vectors u and v. if the vector u and v lie in a plane then the orthogonal vector to both the vectors is orthogonal to the plane as well.

Thus, the cross product of the vectors u and v is orthogonal to the vector u and v. The vectors u and v lie in a plane then the vector $u\times v$ is orthogonal to the plane.