Question

dados os vetores u = 2i - 3j, v= i-j e w= -2i + j , determinar
a) 2u - v
b) v - u + 2w
c) 1/2 u - 2 v - w
d) 3u - 1/2 v - 1/2 w

Answer 1

$\mathbf{u}=2\mathbf{i}-3\mathbf{j}\\ \\ \mathbf{v}=\mathbf{i}-\mathbf{j}\\ \\ \mathbf{w}=-2\mathbf{i}+\mathbf{j}$


a) $2\mathbf{u}-\mathbf{v}$

$=2\cdot \left(2\mathbf{i}-3\mathbf{j} \right )-\left(\mathbf{i}-\mathbf{j} \right )\\ \\ =4\mathbf{i}-6\mathbf{j}-\mathbf{i}+\mathbf{j}\\ \\ =\left(4-1 \right )\cdot \mathbf{i}+\left(-6+1 \right )\cdot \mathbf{j}\\ \\ =3\mathbf{i}-5\mathbf{j}$


b) $\mathbf{v}-\mathbf{u}+2\mathbf{w}$

$\left(\mathbf{i}-\mathbf{j}\right)-\left(2\mathbf{i}-3\mathbf{j} \right )+2\cdot \left(-2\mathbf{i}+\mathbf{j} \right )\\ \\ =\mathbf{i}-\mathbf{j}-2\mathbf{i}+3\mathbf{j} \right )-4\mathbf{i}+2\mathbf{j}\\ \\ =\left(1-2-4 \right )\cdot \mathbf{i}+\left(-1+3+2 \right )\cdot \mathbf{j}\\ \\ =-5\mathbf{i}+4\mathbf{j}$


c) $\dfrac{1}{2}\mathbf{u}-2\mathbf{v}-\mathbf{w}$

$=\dfrac{1}{2}\cdot \left(2\mathbf{i}-3\mathbf{j}\right)-2\cdot \left(\mathbf{i}-\mathbf{j}\right)-\left(-2\mathbf{i}+\mathbf{j}\right)\\ \\ =\mathbf{i}-\dfrac{3}{2}\mathbf{j}-2\mathbf{i}+2\mathbf{j}+2\mathbf{i}-\mathbf{j}\right)\\ \\ =\left(1-2+2 \right )\cdot \mathbf{i}+\left(-\dfrac{3}{2}+2-1 \right )\cdot \mathbf{j}\\ \\ =\mathbf{i}+\dfrac{-3+4-2}{2}\mathbf{j}\\ \\ =\mathbf{i}-\dfrac{1}{2}\mathbf{j}$


d) $3\mathbf{u}-\dfrac{1}{2}\mathbf{v}-\dfrac{1}{2}\mathbf{w}$

$=3\cdot\left(2\mathbf{i}-3\mathbf{j} \right )-\dfrac{1}{2}\cdot \left(\mathbf{i}-\mathbf{j} \right )-\dfrac{1}{2}\cdot\left(-2\mathbf{i}+\mathbf{j} \right )\\ \\ =6\mathbf{i}-9\mathbf{j}-\dfrac{1}{2}\mathbf{i}+\dfrac{1}{2}\mathbf{j}+\mathbf{i}-\dfrac{1}{2}\mathbf{j}\\ \\ =\left(6-\dfrac{1}{2}+1 \right )\cdot \mathbf{i}+\left(-9+\dfrac{1}{2}-\dfrac{1}{2} \right )\cdot \mathbf{j}\\ \\ =\dfrac{12-1+2}{2}\mathbf{i}-9\mathbf{j}\\ \\ =\dfrac{13}{2}\mathbf{i}-9\mathbf{j}$

Answer 2

Dados os vetores u, v e w, determinamos que:

a) 3i - 5j

b) -5i + 4j

c) i - (1/2)j

d) (13/2)i - 9j

Vetores

Vetores são segmentos de retas que possuem módulo ou norma (comprimento), direção e sentido e são usados para descrever grandezas chamadas vetoriais.

Nesta questão, devemos aplicar as seguintes operações com vetores:

• Produto entre vetor e escalar: k·(a, b, c) = (k·a, k·b, k·c);
• Soma de vetores: (a, b, c) ± (d, e, f) = (a±d, b±e, c±f).

Sendo os vetores u = 2i - 3j, v = i - j e w = -2i + j, teremos:

a) 2u - v

= 2·(2i - 3j) - (i - j)

= 4i - 6j - i + j

= 3i - 5j

b) v - u + 2w

= i - j - (2i - 3j) + 2(-2i + j)

= i - j - 2i + 3j - 4i + 2j

= -5i + 4j

c) (1/2)u - 2v - w

= (1/2)(2i - 3j) - 2(i - j) - (-2i + j)

= i - (3/2)j - 2i + 2j + 2i - j

= i - (1/2)j

d) 3u - (1/2)v - (1/2)w

= 3(2i - 3j) - (1/2)(i - j) - (1/2)(-2i + j)

= 6i - 9j - (1/2)i + (1/2)j + i - (1/2)j

= (13/2)i - 9j

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