Anyone good with vectors?/Physics?

Question

A grasshopper makes four jumps. The displacement vectors are (1) 22.0 cm, due west; (2) 27.0 cm, 43.0 ° south of west; (3) 29.0 cm, 59.0 ° south of east; and (4) 26.0 cm, 54.0 ° north of east. Find (a) the magnitude and (b) direction of the resultant displacement. Express the direction as a positive angle with respect to due west.

Answer 1

x = - 22 - 27cos43 + 29cos59 + 26cos54
x = - 11.52803
y = 0 - 27sin43 - 29sin59 + 26sin54
y = - 22.23737
d = 25.04787 @ 62.59741° S of W

Answer 2

Divide each jump into its x and y components and sum the pieces to get the resultant.

x1 = 22cos(180°) = -22
x2 = 27cos(180° + 43°) = 27cos(223°) ≈ -19.74655
x3 = 29cos(360° - 59°) = 29cos(301°) ≈ 14.936104
x4 = 26cos(54°) ≈ 15.282417

x = sum of x's ≈ -11.528029
_________

y1 = 22sin(180°) = 0
y2 = 27sin(180° + 43°) = 27sin(223°) ≈ -18.413956
y3 = 29sin(360° - 59°) = 29sin(301°) ≈ -24.857852
y4 = 26sin(54°) ≈ 21.034442

y = sum of y's ≈ -22.237366

(a) Magnitude of resultant

m = √[(-11.528029)² + (-22.237366)²] ≈ 25.047872

(b) Direction of resultant

tan(θ) = y/x
θ = arctan(y/x)
θ = arctan(-22.237366/-11.528029)
θ = arctan(1.9289825) ≈ 242.597414°

Expressed as a positive direction from due west.

θ = 62.597414° south of west

God bless!

Answer 3

Divide each jump into its x and y components and sum the pieces to get the resultant.

x1 = 22cos(180°) = -22
x2 = 27cos(180° + 43°) = 27cos(223°) ≈ -19.74655
x3 = 29cos(360° - 59°) = 29cos(301°) ≈ 14.936104
x4 = 26cos(54°) ≈ 15.282417

x = sum of x's ≈ -11.528029
_________

y1 = 22sin(180°) = 0
y2 = 27sin(180° + 43°) = 27sin(223°) ≈ -18.413956
y3 = 29sin(360° - 59°) = 29sin(301°) ≈ -24.857852
y4 = 26sin(54°) ≈ 21.034442

y = sum of y's ≈ -22.237366

(a) Magnitude of resultant

m = √[(-11.528029)² + (-22.237366)²] ≈ 25.047872

(b) Direction of resultant

tan(θ) = y/x
θ = arctan(y/x)
θ = arctan(-22.237366/-11.528029)
θ = arctan(1.9289825) ≈ 242.597414°

Expressed as a positive direction from due west.

θ = 62.597414° south of west

Answer 4

x1 = - 22
x2 = - 27 sin 45 = -19.082
x3 = 29 sin 45 = 20.506
x4 = 26 sin 54 = 21.034
. . . . . . . . --------------
. . . sum x = 0.058

y1 = 0
y2 = - 27 cos 45 = -19.082
y3 = - 29 cos 45 = - 20.506
y4 = 26 cos 54 = 15.282
. . . . . . . . --------------
. . . . sum y = - 24.306

magnitude of resultant = sqr(0.058^2 + 24.306^2) = 24.306
tan A = 0.058/24.306
A = 0 deg 8' 12"
angle from west = 180 deg 8' 12"