Add the following vectors using trigonometry (i.e. cosine and sine laws).
9 N on a heading of [S2°W] and 11 N on a heading of [N31°W].
[S2°W] ----> 268°
[N31°W] ---> 121°
Resultant vector
= 9(cos268, sin268) + 11(cos121,sin121)
= (-.03141, -8.9945) + (-5.6654, 9.4288)
= (-5.9795, 0.4343)
magnitude = 5.9953
direction = 175.85°
or
make a sketch, on mine I can use the cosine law to get
R^2 = 9^2 + 11^2 - 2(9)(11)cos33°
= 35.9432...
R = √35.9432... = 5.9953 , same as before.
I will leave it up to you to find the direction angle using the sine law
F=r = 9{182o] + 11[329o].
X = 9*sin182 + 11*sin329 = -5.98 N.
Y = 9*cos182 + 11*Cos329 = 0.4343 N.
Fr = sqrt(X^2 + Y^2) = 5.996 N. = Resultant force.
TanA = X/Y, A = -85.9o = 85.9o W. of N. = 274.1 CW from +Y-axis = 175.9o CCW from +X-axis
To add the given vectors using trigonometry, we will first convert the headings into components using the cosine and sine laws.
Vector 1: 9 N on a heading of [S2°W]
To find the components, let's assume a coordinate system where North is the positive y-axis and East is the positive x-axis.
The given direction is South 2 degrees West, which means it is in the fourth quadrant.
To find the x-component:
cos(180 - 2) = cos(178) ≈ -0.9994
x₁ = 9 N * -0.9994 ≈ -8.9946 N
To find the y-component, multiply the magnitude (9 N) by the sine of the angle:
sin(180 - 2) = sin(178) ≈ 0.0349
y₁ = 9 N * 0.0349 ≈ 0.3141 N
Vector 2: 11 N on a heading of [N31°W]
The given direction is North 31 degrees West, which means it is in the second quadrant.
To find the x-component:
cos(180 - 31) = cos(149) ≈ -0.5299
x₂ = 11 N * -0.5299 ≈ -5.829 N
To find the y-component, multiply the magnitude (11 N) by the sine of the angle:
sin(180 - 31) = sin(149) ≈ 0.8480
y₂ = 11 N * 0.8480 ≈ 9.328 N
Now, we can add the x-components and y-components separately to find the resulting sum:
X-component: -8.9946 N + (-5.829 N) = -14.8236 N
Y-component: 0.3141 N + 9.328 N = 9.6421 N
Therefore, the resulting vector sum is approximately -14.8236 N in the x-direction and 9.6421 N in the y-direction.
To add vectors using trigonometry, we need to break down each vector into its horizontal and vertical components.
Let's start with the first vector, 9 N on a heading of S2°W.
Step 1: Determine the horizontal and vertical components of the vector.
We can use trigonometric functions to determine the horizontal and vertical components.
The vertical component can be calculated using the sine law:
Vertical component = magnitude × sin(angle)
Vertical component = 9 N × sin(2°)
Vertical component = 9 N × sin(2°)
Vertical component ≈ 0.314 N (rounded to three decimal places)
Similarly, the horizontal component can be calculated using the cosine law:
Horizontal component = magnitude × cos(angle)
Horizontal component = 9 N × cos(2°)
Horizontal component = 9 N × cos(2°)
Horizontal component ≈ 8.998 N (rounded to three decimal places)
Therefore, the first vector can be expressed as (8.998 N, 0.314 N).
Now, let's move on to the second vector, 11 N on a heading of N31°W.
Step 1: Determine the horizontal and vertical components of the vector.
Just like in the previous step, we can use trigonometric functions to determine the horizontal and vertical components.
Vertical component = 11 N × sin(31°)
Vertical component ≈ 5.623 N (rounded to three decimal places)
Horizontal component = 11 N × cos(31°)
Horizontal component ≈ 9.430 N (rounded to three decimal places)
Therefore, the second vector can be expressed as (9.430 N, 5.623 N).
To find the sum of these vectors, we simply add the horizontal and vertical components separately.
Sum of the horizontal components = 8.998 N + 9.430 N ≈ 18.428 N (rounded to three decimal places)
Sum of the vertical components = 0.314 N + 5.623 N ≈ 5.937 N (rounded to three decimal places)
Therefore, the sum of the vectors is approximately (18.428 N, 5.937 N) when rounded to three decimal places.