-9 Dois vetores são dados por
ll & a=(4,0 m) i-(3,0 m) j+(1,0 m) k e & b=(-1,0 m) i+(1,0 m) j+(4,0 m) k
Determine, na notação dos vetores unitários, (a) a+b; (b) a-b; (c) um terceiro vetor, c, tal que a-b +c=0.

Passo 1

O enunciado dá dois vetores em R^3
a=(4,0 m),imath-(3,0 m),jmath+(1,0 m),k
b=(-1,0 m),imath+(1,0 m),jmath+(4,0 m),k
e pede (a) a+b, (b) a-b e (c) um vetor c tal que a-b+c=0. Vamos usar adição e subtração de vetores componte a componente.

Passo 2

Para somar vetores, somamos cada componente correspondente:
a+b =(4,0 m)+(-1,0 m);imath +[-3,0 m+1,0 m];jmath +(1,0 m+4,0 m);k
Logo
a+b=(3,0 m),imath-(2,0 m),jmath+(5,0 m),k

Passo 3

Para subtrair vec b de vec a, subtraímos componente a componente:
a-b =(4,0 m-(-1,0 m)),imath +[-3,0 m-1,0 m],jmath +(1,0 m-4,0 m),k
Logo
a-b=(5,0 m),imath-(4,0 m),jmath-(3,0 m),k

Passo 4

Queremos c tal que a-b+c=0. Basta escolher
c=-(a-b)
ou seja, invertemos o sinal de cada componente de a-b:
c=-(5,0 m),imath + (4,0 m),jmath + (3,0 m),k

Resposta

(a) a+b=(3,0 m),imath-(2,0 m),jmath+(5,0 m),k
(b) a-b=(5,0 m),imath-(4,0 m),jmath-(3,0 m),k
(c) c=-(5,0 m),imath+(4,0 m),jmath+(3,0 m),k

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