Question

Add the following two vectors if \vec{A} + \vec{B} = \vec{C} Keep a few digits. |\vec{A}| = 10N \theta_{A} = 30^{\circ} |\vec{B}| = 8N \theta_{B} = 10^{\circ}

Answer 1

The addition of vectors allows us to find that the resulting vector is

         c = 17.73 N  with an angle of θ = 21.1º

Giving parameters

• The vector A a = 10 N  θ = 30º

• The vector B B = 8 N   θ = 10º

To find

• the vector adds

Vector magnitudes have magnitude and direction, so their addition must be done using the vector aliebre.

 

Analytical method is one of the easiest ways to use vectors, it consists of the following parts:

• Decompose vectors into a coordinate system

• Perform the algebraic sum of the components

• Find the resultant vector.

Let's decompose the two vectors into a coordinate system with the horizontal x axis and the vertical y axis, in the attachment we can see a diagram of the vectors

Vector A

            cos 30 = $\frac{A_x}{A}$

            sin 30 = $\frac{A_y}{A}$

            Aₓ = A cos 30

            $A_y$ = A sin 30

            Aₓ = 10 cos 30

           A_y = 10 sin 30

            Aₓ = 8.66 N

            A_y = 5.00 N

Vector B

            cos 10 = $\frac{B_x}{B}$

            sin 10 = $\frac{B_y}{B}$

            Bₓ = B cos 10

            $B_y$ = B sin 10

            Bₓ = 8 cos 10 = 7.878 N

            B_y = 8 sin 10 = 1.389 N

We perform the algebraic sum

x-axis

            x = Aₓ + B_y

            x = 8.66 +7.878

            x = 16.54 N

y-axis

           y = A_y + B_y

           y = 5.00 + 1.389

           y = 6.39 N

We construct the resulting vector

For the module we use the Pythagoras' theorem

           c = $\sqrt{x^2+y^2}$

           c =$\sqrt{16.54^2 + 6.39^2}$  

         c = 17.73 N

For the direction we use trigonometry

         tan θ =  $\frac{y}{x}$  

         θ = tan⁻¹  $\frac{y}{x}$

         θ = tan⁻¹  $\frac{6.39} { 16.54}$

         θ = 21.1º

measured counterclockwise from the positive side of the x-axis

In conclusion using vector addition we can find the resulting vector is

         c = 17.73 N     with an angle of θ = 21.1º

Learn more about vector addition here:

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