Adding Vectors Algebraically

The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. While this method is perhaps not as ‘entertaining’ as either of the others, it is (by far) the most accurate and easiest in the long run.

This method is based on the fact that any given vector can be separated into components that act at right angles to each other.

This method is by far (after you get used to it) the easiest and most reliable method for finding the resultant of any number of vector quantities.

Most textbooks use the cosine law for this method, but a quick, much easier method is to use the trig functions of each vector to separate it into x and y (vertical and horizontal) components, then combine the components.

Summary of equations:

x = r cos 

y = r sin 

x² + y² = r²

 = tan^-1(y/x)

 

EXAMPLE: the following forces act simultaneously on an object

V₁ = 500 N force acts at an angle of 17^o W of N

V₂ = 315 N at 27^o W of S

For the first vector:

Step1:   reference the angle back to due E
17^o W of N = 90^o + 17^o from due E = 107^o

Step 2:   now use the cos function to find the x (horizontal) component ( x = r cos)
cos 107^o X 500 N = -146.186 N

Step 3:   use the sin function to find the y (vertical) component (y = r sin )
sin 107^o X 500 N = 478.152 N

For the second vector:

Step1:    reference angle back to due E
27^o W of S = 270^o (due S) 27^o (back west) from due E = 243^o

Step 2:   use cos function to find x component
cos 243^o X 315 N = -143.007 N

Step 3:   use sin function to find y component
sin 243^o X 315 N = -280.667 N

• Notice that using this method, reference everything back to due E, means your calculator automatically gives you pos/neg, which is important when you combine the components!
• Now, combine the x components
  -146.186 + (-143.007) = -289.193 N
• Combine the y components
  478.152 + (-280.667) = 197.485 N
• Look at the signs of the resultant x and y: x is neg and y is pos, so your resultant will be in the 2^nd quad; this means the angle of the resultant will be (using this method) N of due W
• Now use the Pythagorean Theorem to find the resultant (by separating the vectors into x and y components, you are forcing the resultant to be the hypotenuse of a right triangle).

Resultant² = x² + y²

= (-289.193)² + (197.485)²

= sq rt (122632.9165)

resultant = 350.190 N

using sig fig (2 because your angles had 2 digits)

resultant = 350 N or 3.5×102 N

Now find the orientation (angle)

Tan angle (the angle of the resultant above due W) = y/x          [tan^-1 (y/x)]
= 197.485/289.193

NOTE: you don’t need to use pos/neg here since you know its in the 2^nd quad and both are negative

tan  ^-1 = 0.682883057
(use inv tan)
angle = 34^o

So your resultant = 3.4 X 10² N 34^o N of W

• Now go to Physics Classroom trig method on the following website and read through that explanation as reinforcement.   Use the Components of a Vector widget at the bottom of the page to resolve vectors into their components.

 

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