1. Add the vectors A, B, and C shown in the figure using the component method.
A = 5.0m, B = 7.0m, and C = 4.0m
Adding like components of two or more vectors gives the corresponding component of the resultant vector. The magnitude and direction of resultant vector can be easily determined by simple trigonometry we learned in problem (3).
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2. To add vectors in this problem we need to;
Add the vectors A, B, and C shown in the figure using the component method.
A = 5.0m, B = 7.0m, and C = 4.0m
Solution:
Here we learn another way to add two or more vectors. This method is called component method.
To add vectors in this problem we need to;
Calculate the components of each vector
Add like components to calculate corresponded component of resultant vector
Rx = Ax + Bx + Cx Ry = Ay + By + Cy
3. Then determine the magnitude and direction of R
R2 = Rx2 + Ry2 and tg θ = Ry / Rx
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3. Step 1: Calculate components for each vector
Add the vectors A, B, and C shown in the figure using the component method.
A = 5.0m, B = 7.0m, and C = 4.0m
Now we start step by step
Step 1: Calculate components for each vector
Ax =
Bx =
Cx =
Ay =
By =
Cy =
A Cos 20⁰ = + (5.0m) Cos 20⁰  Ax = + 4.7m
If a component of a vector points along negative part of axis, its sign will be negative.
- B Cos 40⁰ = - (7.0m) Cos 40⁰  Bx = -5.4m
+C sin 25⁰ = + (4.0) Sin 25⁰  Cx = + 1.7m
Projection of “C” along “X’ axis is Cx = C Cos θ whereas θ is the angle it makes with “X” axis. So we can write:
Cx = C Cos (90-25) = C Cos (65⁰) and we know Cos (90-α) = Sin (α)  Cos 65⁰ = Sin25⁰
+A Sin 20⁰ = + (5.0m) Sin 20⁰  Ay = +1.7m
+B Sin 40⁰ = + (7.0m) Sin 40⁰  By = + 4.5m
- C Cos 25⁰ = - (4.0) Cos 25⁰  Cy= - 3.6m
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4. Step 2: Adding Like Components
Add the vectors A, B, and C shown in the figure using the component method.
A = 5.0m, B = 7.0m, and C = 4.0m
Step 2: Adding Like Components
Rx =
Ry =
Ax + Bx + Cx = +4.7m + (-5.4m) + 1.7m  Rx = + 1.0m
Positive sign shows that the components of the resultant vector point along +x and +y axes.
Ay + By + Cy = +1.7m +4.5m + (-3.6m)  Ry = + 2.6m
Step 3: Calculating the Magnitude and Direction of Vector
R2 =
tg θ =
Rx2 + Ry2 = (1)2 + (2.6)2 → R= 2.8m
Magnitude of the resultant vector
Ry Rx or θ = Arc tg Ry Rx = Arc tg ( 2.6m 1.0m ) → θ = 69⁰
Resultant vector makes an angle of 69⁰ with +x axis.
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5. So we have the resultant vector shown as below:
Add the vectors A, B, and C shown in the figure using the component method.
A = 5.0m, B = 7.0m, and C = 4.0m
So we have the resultant vector shown as below:
A = 5.0m Ax = +4.7m Ay = +1.7m
B = 7.0m Bx = -5.4m By = + 4.5m
C = 4.0m Cx = + 1.7m Cy = - 3.6m
R= 2.8m Rx = + 1.0m Ry = + 2.6m
Ry = Ay + By + Cy
Rx = Ax + Bx + Cx
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