15. Sejam u=(-3,2,1,0),v=(4,7,-3,2) e w=(5,-2,8,1) Encontre os componentes de (a) V-W (b) 2u+7v (c) -u+(v-4w) (d) 6(u-3v) (e) -V-W (f) (6v-w)-(4u+v)

Vamos corrigir e detalhar cada parte da questão:<br /><br />(a) \( V - W \)<br /><br />\[ V - W = (4, 7, -3, 2) - (5, -2, 8, 1) \]<br />\[ = (4 - 5, 7 - (-2), -3 - 8, 2 - 1) \]<br />\[ = (-1, 9, -11, 1) \]<br /><br />(b) \( 2u + 7v \)<br /><br />\[ 2u + 7v = 2(-3, 2, 1, 0) + 7(4, 7, -3, 2) \]<br />\[ = (-6, 4, 2, 0) + (28, 49, -21, 14) \]<br />\[ = (-6 + 28, 4 + 49, 2 - 21, 0 + 14) \]<br />\[ = (22, 53, -19, 14) \]<br /><br />(c) \( -u + (v - 4w) \)<br /><br />\[ -u + (v - 4w) = (-3, 2, 1, 0) + ((4, 7, -3, 2) - 4(5, -2, 8, 1)) \]<br />\[ = (-3, 2, 1, 0) + (4 - 20, 7 - (-8), -3 - 32, 2 - 4) \]<br />\[ = (-3, 2, 1, 0) + (-16, 15, -35, -2) \]<br />\[ = (-3 - 16, 2 + 15, 1 - 35, 0 - 2) \]<br />\[ = (-19, 17, -34, -2) \]<br /><br />(d) \( 6(u - 3v) \)<br /><br />\[ 6(u - 3v) = 6((-3, 2, 1, 0) - 3(4, 7, -3, 2)) \]<br />\[ = 6((-3, 2, 1, 0) - (12, 21, -9, 6)) \]<br />\[ = 6((-3 - 12, 2 - 21, 1 + 9, 0 - 6)) \]<br />\[ = 6((-15, -19, 10, -6)) \]<br />\[ = (-90, -114, 60, -36) \]<br /><br />(e) \( -V - W \)<br /><br />\[ -V - W = - (4, 7, -3, 2) - (5, -2, 8, 1) \]<br />\[ = (-4, -7, 3, -2) - (5, -2, 8, 1) \]<br />\[ = (-4 - 5, -7 - (-2), 3 - 8, -2 - 1) \]<br />\[ = (-9, -5, -5, -3) \]<br /><br />(f) \( (6v - w) - (4u + v) \)<br /><br />\[ (6v - w) - (4u + v) = (6(4, 7, -3, 2) - (5, -2, 8, 1)) - (4(-3, 2, 1, 0) + (4, 7, -3, 2)) \]<br />\[ = (24, 42, -18, 12) - (5, -2, 8, 1) - (-12, 8, 4, 4) - (4, 7, -3, 2) \]<br />\[ = (24 - 5, 42 - (-2), -18 - 8, 12 - 1) - (-12 - 4, 8 - 7, 4 - (-3), 4 - 2) \]<br />\[ = (19, 44, -26, 11) - (-16, 1, 7, 2) \]<br />\[ = (19 + 16, 44 - 1, -26 - 7, 11 - 2) \]<br />\[ = (35, 43, -33, 9) \]<br /><br />Portanto, as respostas corrigidas são:<br /><br />(a) \( (-1, 9, -11, 1) \)<br /><br />(b) \( (22, 53, -19, 14) \)<br /><br />(c) \( (-19, 17, -34, -2) \)<br /><br />(d) \( (-90, -114, 60, -36) \)<br /><br />(e)