Vektor dalam R^3 .Diketahui p=(-3 2 6), q=(6 -1 6), dan

a. $-p = (3, -2, -6)$, $|-p|=7$; b. $p+2q = (9, 0, 18)$, $|p+2q|=9\sqrt{5}$; c. $3r-2p+2q = (24, -6, -9)$, $|3r-2p+2q|=3\sqrt{77}$

Pembahasan

Diketahui vektor $p=(-3, 2, 6)$, $q=(6, -1, 6)$, dan $r=(2, 0, -3)$. Kita akan menentukan vektor-vektor yang diminta dan menghitung besarnya.

a. Vektor $-p$
$-p = -(-3, 2, 6) = (3, -2, -6)$
Besar vektor $-p$, $|-p| = \sqrt{3^2 + (-2)^2 + (-6)^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$

b. Vektor $p + 2q$
$2q = 2(6, -1, 6) = (12, -2, 12)$
$p + 2q = (-3, 2, 6) + (12, -2, 12) = (-3+12, 2-2, 6+12) = (9, 0, 18)$
Besar vektor $p+2q$, $|p+2q| = \sqrt{9^2 + 0^2 + 18^2} = \sqrt{81 + 0 + 324} = \sqrt{405} = \sqrt{81 \times 5} = 9\sqrt{5}$

c. Vektor $3r - 2p + 2q$
$3r = 3(2, 0, -3) = (6, 0, -9)$
$-2p = -2(-3, 2, 6) = (6, -4, -12)$
$2q = 2(6, -1, 6) = (12, -2, 12)$
$3r - 2p + 2q = (6, 0, -9) + (6, -4, -12) + (12, -2, 12)$
$3r - 2p + 2q = (6+6+12, 0-4-2, -9-12+12) = (24, -6, -9)$
Besar vektor $3r-2p+2q$, $|3r-2p+2q| = \sqrt{24^2 + (-6)^2 + (-9)^2} = \sqrt{576 + 36 + 81} = \sqrt{693}$
$\sqrt{693} = \sqrt{9 \times 77} = 3\sqrt{77}$