Chapter 11.3, Problem 18E

To determine

The value of the expression $u\text{×}v$ and show that the product obtained is orthogonal to both u and v.

Answer to Problem 18E

The value of the expression $\text{u×v}$ is $-\text{34}i\text{ + 45}j-52k$ and both the vectors u and v are orthogonal to $u\text{×}v$.

Explanation of Solution

Given information:

The value of the vectors in component form $u=\langle 6,8,3\rangle$ and $v=\langle 5,-2,-5\rangle$ .

Formula used:

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 ,

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

Any two non-zero vectors are said to be orthogonal to each other if and only if their dot product is zero, i.e. the angle between the two vectors is ${90}^{\circ }$ . Dot product of any two vectors, say, $x=x_{1}i+x_{2}j+x_{3}k$ and $y=y_{1}i+y_{2}j+y_{3}k$ is expressed as,

  $\begin{array}{c}x\cdot y=\left(x_{1}i+x_{2}j+x_{3}k\right)\cdot \left(y_{1}i+y_{2}j+y_{3}k\right)\\ =x_1{y}_1+x_2{y}_2+x_3{y}_3\end{array}$

Calculation:

Consider the given value of the vectors $u=\langle 6,8,3\rangle$ and $v=\langle 5,-2,-5\rangle$ .

Recall that ifthere 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 ,

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

Apply it,

So, the cross product of u and v vectors will be calculated as,

  $\begin{array}{c}u\times v=\begin{array}{ccc}i& j& k\\ 6& 8& 3\\ 5& -2& -5\end{array}\\ \text{= }\left(\left[\left(-5\right)\times 8\right]-\left[3\times \left(-2\right)\right]\right)i-\left(\left[6\times \left(-5\right)\right]-\left[5\times 3\right]\right)j+\left(\left[6\times \left(-2\right)\right]-\left[8\times 5\right]\right)k\\ \text{=}\left(-40-\left(-6\right)\right)i-\left(-30-15\right)j+\left(-12-40\right)k\\ \text{=}\left(-40+6\right)i-\left(-45\right)j+\left(-52\right)k\end{array}$

Simplify it further as,

  $\begin{array}{c}u\times v=\left(-40+6\right)i-\left(-45\right)j+\left(-52\right)k\\ \text{=}-34i+45j-52k\end{array}$

Now, recall that any two non-zero vectors are said to be orthogonal to each other if and only if their dot product is zero, i.e. the angle between the two vectors is ${90}^{\circ }$ . Dot product of any two vectors, say, $x=x_{1}i+x_{2}j+x_{3}k$ and $y=y_{1}i+y_{2}j+y_{3}k$ is expressed as,

  $\begin{array}{c}x\cdot y=\left(x_{1}i+x_{2}j+x_{3}k\right)\cdot \left(y_{1}i+y_{2}j+y_{3}k\right)\\ =x_1{y}_1+x_2{y}_2+x_3{y}_3\end{array}$

So, to prove that the product obtained is orthogonal to both u and v, we will calculate the dot product of u and v with $u\text{×}v$ .

So, the dot product of u and $u\text{×}v$ will be calculated as,

  $\begin{array}{c}u\cdot \left(u\times v\right)=\left(\text{6}i\text{+8}j\text{+3B}\right)\cdot \left(-\text{34}i\text{+45}j-52k\right)\\ =\left[6\cdot \left(-34\right)\right]+\left[45\cdot 8\right]+\left[3\cdot \left(-52\right)\right]\\ =-204+360-156\\ =0\end{array}$

Since, dot product of u and $u\text{×}v$ is zero, so, u is orthogonal to $u\text{×}v$ .

Now, dot product of v and $u\text{×}v$ will be calculated as,

  $\begin{array}{c}v\cdot \left(u\times v\right)=\left(\text{5}i-2j-\text{5}k\right)\cdot \left(-\text{34}i\text{+ 45}j-52k\right)\\ =\left[5\cdot \left(-34\right)\right]+\left[45\cdot \left(-2\right)\right]+\left[\left(-5\right)\cdot \left(-52\right)\right]\\ =-170-90+260\\ =0\end{array}$

Since, dot product of v and $u\text{×}v$ is zero, so, v is also orthogonal to $u\text{×}v$ .

Hence, the value of the expression $\text{u×v}$ is $-\text{34}i\text{ + 45}j-52k$ and it is also proved that both u and v are orthogonal to $u\text{×}v$ .