Q. 25 - Calculus

Question

In Exercises 22–29 compute the indicated quantities when $u=(2,1,-3), v=(4,0,1), \mathrm{and} w=(-2,6,5)$

$(u\times v)\times w \mathrm{and} u\times (v\times w)$

Step-by-Step Solution

Answer

The value of $(u\times v)\times w=94i-3j+34k$ and $u\times (v\times w)=90i-30j+50k$.

Step 1. Given Information

In Exercises 22–29 compute the indicated quantities when$u=(2,1,-3), v=(4,0,1), \mathrm{and} w=(-2,6,5)$

We have to find the value of $(u\times v)\times w \mathrm{and} u\times (v\times w)$

Step 2. Firstly finding the value of ( u × v ) × w

The cross product of $u\times v=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& -3\\ 4& 0& 1\end{array}\right]$

Now solving the cross product.

$$\begin{gathered} u\times v=\left(\right(1\left)\right(1)-(-3\left)\right(0\left)\right)i+\left(\right(2\left)\right(1)-(-3\left)\right(4\left)\right)j+\left(\right(2\left)\right(0)-(1\left)\right(4\left)\right)k \\ u\times v=(1-0)i+(2+12)j+(0-4)k \\ u\times v=1i+14j-4k \end{gathered}$$

Step 3. Now the vectors of u × v   is   ( 1 , 14 , - 4 ) .

Now finding the value of $(u\times v)\times w$

$$\begin{gathered} (u\times v)\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 14& -4\\ -2& 6& 5\end{array}\right] \\ (u\times v)\times w=\left(\right(14\left)\right(5)-(-4\left)\right(6\left)\right)i+\left(\right(1\left)\right(5)-(-4\left)\right(-2\left)\right)j+\left(\right(1\left)\right(6)-(14\left)\right(-2\left)\right)k \\ (u\times v)\times w=(70+24)i+(5-8)j+(6+28)k \\ (u\times v)\times w=94i-3j+34k \end{gathered}$$

Step 4. Firstly finding the value of v × w

The cross product of $v\times w=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 4& 0& 1\\ -2& 6& 5\end{array}\right]$

$$\begin{gathered} v\times w=\left(\right(0\left)\right(5)-(1\left)\right(6\left)\right)i+\left(\right(4\left)\right(5)-(1\left)\right(-2\left)\right)j+\left(\right(4\left)\right(6)-(0\left)\right(-2\left)\right)k \\ v\times w=(0-6)i+(20+2)j+(24+0)k \\ v\times w=-6i+22j+24k \end{gathered}$$

Step 5. Now the vectors of v × w   is   ( - 6 , 22 , 24 )

$$\begin{gathered} u\times (v\times w)=\det \left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 2& 1& -3\\ -6& 22& 24\end{array}\right] \\ u\times (v\times w)=\left(\right(1\left)\right(24)-(-3\left)\right(22\left)\right)i+\left(\right(2\left)\right(24)-(-3\left)\right(-6\left)\right)j+\left(\right(2\left)\right(22)-(1\left)\right(-6\left)\right)k \\ u\times (v\times w)=(24+66)i+(48-18)j+(44+6)k \\ u\times (v\times w)=90i-30j+50k \end{gathered}$$