Respuesta :
Answer:
18ij-28kl+46lj
Step-by-step explanation:
A vector is a quantity which has both magnitude and direction.
For vectors u and v, vector perpendicular to both u and v is equal to u×v
Given:
[tex]v= [-2,-8,-7,2]\\u= [6,7,-2,8][/tex]
Solution:
[tex]u\times v=\left | \begin{matrix}i&j&k&l\\-2&-8&-7&2\\6&7&-2&8 \end{matrix} \right |\\=i\left | \begin{matrix}j&k&l\\-8&-7&2\\7&-2&8 \end{matrix} \right |-j\left | \begin{matrix}i&k&l\\-2&-7&2\\6&7&8 \end{matrix} \right |+k\left | \begin{matrix}i&j&l\\-2&-8&2\\6&7&8 \end{matrix} \right |-l\left | \begin{matrix}i&j&k\\-2&-8&-7\\6&7&-2 \end{matrix} \right |[/tex]
Solving: [tex]i\left | \begin{matrix}j&k&l\\-8&-7&2\\7&-2&8 \end{matrix} \right |[/tex]
[tex]i\left | \begin{matrix}j&k&l\\-8&-7&2\\7&-2&8 \end{matrix} \right |=i\left [ j(-56+4)-k(-64-14)+l(16+49) \right ]=i(-52j+78k+65l)[/tex]
Solving: [tex]-j\left | \begin{matrix}i&k&l\\-2&-7&2\\6&7&8 \end{matrix} \right |[/tex]
[tex]-j\left | \begin{matrix}i&k&l\\-2&-7&2\\6&7&8 \end{matrix} \right |=-j\left [ i(-56-14)-k(-16-12)+l(-14+42) \right ]=j(70i-28k-28l)[/tex]
Solving: [tex]k\left | \begin{matrix}i&j&l\\-2&-8&2\\6&7&8 \end{matrix} \right |[/tex]
[tex]k\left | \begin{matrix}i&j&l\\-2&-8&2\\6&7&8 \end{matrix} \right |=k\left [ i(-64-14)-j(-16-12)+l(-14+48) \right ]=k(-78i+28j+34l)[/tex]
Solving: [tex]-l\left | \begin{matrix}i&j&k\\-2&-8&-7\\6&7&-2 \end{matrix} \right |[/tex]
[tex]-l\left | \begin{matrix}i&j&k\\-2&-8&-7\\6&7&-2 \end{matrix} \right |=-l\left [ i(16+49)-j(4+42)+k(-14+48) \right ]=-l(65i-46j+34k)[/tex]
Therefore,
u×v = [tex]i(-52j+78k+65l)+j(70i-28k-28l)+k(-78i+28j+34l)-l(65i-46j+34k)\\=-52ij+78ik+65il+70ij-28jk-28kl-78ik+28jk+34kl-65li+46lj-34kl\\=18ij-28kl+46lj[/tex]
So, [tex]18ij-28kl+46lj[/tex] is perpendicular to both u and v
