diketahui vektor a=i+2j-xk vektor b=3i-2j+k dan c=2i+j+2k. jika a tegak lurus c maka (a+b).(a-c) hasilnya adalah

Kode : 12.2.4 [Kelas 12 Matematika BAB 4 - Vektor]

Diketahui
[tex]a= \left[\begin{array}{ccc}1\\2\\-x\end{array}\right] \\b= \left[\begin{array}{ccc}3\\-2\\1\end{array}\right] \\c = \left[\begin{array}{ccc}2\\1\\2\end{array}\right][/tex]

Karena saling tegak lurus, [tex]a.c=0[/tex]
Sehingga
[tex] \left[\begin{array}{ccc}1\\2\\-x\end{array}\right] . \left[\begin{array}{ccc}2\\1\\2\end{array}\right]=0[/tex]
(1)(2) + (2)(1) + (-x)(2) = 0
4 = 2x
x = 2
Jadi vektor [tex]a= \left[\begin{array}{ccc}1\\2\\-2\end{array}\right][/tex]

Siapkan a + b
[tex]a+b= \left[\begin{array}{ccc}1\\2\\-2\end{array}\right]+ \left[\begin{array}{ccc}3\\-2\\1\end{array}\right] = \left[\begin{array}{ccc}4\\0\\-1\end{array}\right] [/tex]
Siapkan a - c
[tex]a-c= \left[\begin{array}{ccc}1\\2\\-2\end{array}\right]- \left[\begin{array}{ccc}2\\1\\2\end{array}\right] = \left[\begin{array}{ccc}-1\\1\\-4\end{array}\right] [/tex]

(a + b).(a - c) = ? 

[tex] \left[\begin{array}{ccc}4\\0\\-1\end{array}\right] . \left[\begin{array}{ccc}-1\\1\\-4\end{array}\right] =(4)(-1)+(0)(1)+(-1)(-4) [/tex]

Jadi (a + b).(a - c) = 0.