Vectors v and w are given in magnitude and direction form. find the coordinate representation of the sum v + w
and the difference v − w. give coordinates to the nearest tenth of a unit.
a. v: magnitude 12, direction 50° east of north
w: magnitude 8, direction 30° north of east
b. v: magnitude 20, direction 54° south of east
w: magnitude 30, direction 18° west of south

a) v + w = 13.2 i + 14.6 j

v - w = 5.2 i + 0.8 j

b) v + w = 2.5 i - 44.7 i

v - w = 21 i + 12.4 i

Step-by-step explanation:

a) For v:  ∡ = 90º-50º = 40º

v = 12*cos 40º i + 12*Sin40º j

For w:  ∡ = 90º-30º = 60º

w = 8*Cos 60º i + 8*Sin 60º j

v + w = (12*cos 40º+8*Cos 60º) i + (12*Sin40º+8*Sin 60º) j

v + w = 13.2 i + 14.6 j

v - w = (12*cos 40º-8*Cos 60º) i + (12*Sin40º-8*Sin 60º) j

v - w = 5.2 i + 0.8 j

b) For v:  ∡ = -54º (clockwise)  

v = 20*cos (-54º) i + 20*Sin(-54º) j

For w:  ∡ = 270º-18º = 252º

w = 30*Cos 252º i + 30*Sin 252º j

v + w = (20*cos (-54º)+30*Cos 252º) i + (20*Sin(-54º)+30*Sin 252º) j

v + w = 2.5 i - 44.7 i

v - w = (20*cos (-54º)-30*Cos 252º) i + (20*Sin(-54º)-30*Sin 252º) j

v - w = 21 i + 12.4 i

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

answer: options

step-by-step explanation: