Question

if A= i+j+2k, B= 2i+j+k, C= i-2j-2k. find the magnitude and direction cos of. 1. (A+B+C). 2. (A-B+C)

Answer 1

Let $A = (1,1,2), B = (2,1,1), C = (1,-2,-2)$

1) Adding the components, obtain:

$$\mathbf{v}_1 =\mathbf{A} + \mathbf{B} + \mathbf{C} = (1+2+1, 1+1-2,2+1-2) = (4,0,1)$$

The magnitude is:

$$|\mathbf{v}_1| = \sqrt{4^2 + 0^2 + 1^2} = \sqrt{17}$$

The direction cos are:

$$\cos a = \dfrac{4}{\sqrt{17}}\\ \cos b = \dfrac{0}{\sqrt{17}} = 0\\ \cos c = \dfrac{1}{\sqrt{17}}$$

2) Adding the components, obtain:

$$\mathbf{v}_2 =\mathbf{A} - \mathbf{B} + \mathbf{C} = (1-2+1, 1-1-2,2-1-2) = (0,-2,-1)$$

The magnitude is:

$$|\mathbf{v}_1| = \sqrt{0^2 + (-2)^2 + (-1)^2} = \sqrt{5}$$

The direction cos are:

$$\cos a = \dfrac{0}{\sqrt{5}} = 0\\ \cos b = -\dfrac{2}{\sqrt{5}}\\ \cos c = -\dfrac{1}{\sqrt{5}}$$