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  • Understand the rules of vector addition and subtraction using analytical methods.
  • Apply analytical methods to determine vertical and horizontal component vectors.
  • Apply analytical methods to determine the magnitude and direction of a resultant vector.

Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.

Resolving a vector into perpendicular components

Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like A size 12{A} {} in [link] , we may wish to find which two perpendicular vectors, A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} , add to produce it.

In the given figure a dotted vector A sub x is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A sub y at an angle theta from the x axis. On the graph a vector A, inclined at an angle theta with x axis is shown. Therefore vector A is the sum of the vectors A sub x and A sub y.
The vector A size 12{A} {} , with its tail at the origin of an x , y -coordinate system, is shown together with its x - and y -components, A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} . These vectors form a right triangle. The analytical relationships among these vectors are summarized below.

A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} are defined to be the components of A size 12{A} {} along the x - and y -axes. The three vectors A size 12{A} {} , A x size 12{A rSub { size 8{x} } } {} , and A y size 12{A rSub { size 8{y} } } {} form a right triangle:

A x  + A y  = A . size 12{A rSub { size 8{x} } bold " + A" rSub { size 8{y} } bold " = A."} {}

Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if A x = 3 m size 12{A rSub { size 8{x} } } {} east, A y = 4 m size 12{A rSub { size 8{y} } } {} north, and A = 5 m size 12{A} {} north-east, then it is true that the vectors A x  + A y  = A size 12{A rSub { size 8{x} } bold " + A" rSub { size 8{y} } bold " = A"} {} . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,

3 m + 4 m   5 m alignl { stack { size 12{"3 M + 4 M "<>" 5 M"} {} # {}} } {}

Thus,

A x + A y A size 12{A rSub { size 8{x} } +A rSub { size 8{y} }<>A} {}

If the vector A size 12{A} {} is known, then its magnitude A size 12{A} {} (its length) and its angle θ size 12{θ} {} (its direction) are known. To find A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} , its x - and y -components, we use the following relationships for a right triangle.

A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {}

and

A y = A sin θ . size 12{A rSub { size 8{y} } =A"sin"θ"."} {}
]A dotted vector A sub x whose magnitude is equal to A cosine theta is drawn from the origin along the x axis. From the head of the vector A sub x another vector A sub y whose magnitude is equal to A sine theta is drawn in the upward direction. Their resultant vector A is drawn from the tail of the vector A sub x to the head of the vector A-y at an angle theta from the x axis. Therefore vector A is the sum of the vectors A sub x and A sub y.
The magnitudes of the vector components A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} can be related to the resultant vector A size 12{A} {} and the angle θ size 12{θ} {} with trigonometric identities. Here we see that A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {} and A y = A sin θ size 12{A rSub { size 8{y} } =A"sin"θ} {} .

Suppose, for example, that A size 12{A} {} is the vector representing the total displacement of the person walking in a city considered in Kinematics in Two Dimensions: An Introduction and Vector Addition and Subtraction: Graphical Methods .

In the given figure a vector A of magnitude ten point three blocks is inclined at an angle twenty nine point one degrees to the positive x axis. The horizontal component A sub x of vector A is equal to A cosine theta which is equal to ten point three blocks multiplied to cosine twenty nine point one degrees which is equal to nine blocks east. Also the vertical component A sub y of vector A is equal to A sin theta is equal to ten point three blocks multiplied to sine twenty nine point one degrees,  which is equal to five point zero blocks north.
We can use the relationships A x = A cos θ size 12{A rSub { size 8{x} } =A"cos"θ} {} and A y = A sin θ size 12{A rSub { size 8{y} } =A"sin"θ} {} to determine the magnitude of the horizontal and vertical component vectors in this example.

Then A = 10.3 size 12{A} {} blocks and θ = 29.1º size 12{"29.1º"} , so that

A x = A cos θ = ( 10.3 blocks ) ( cos 29.1º ) = 9.0 blocks size 12{}
A y = A sin θ = ( 10.3 blocks ) ( sin 29.1º ) = 5.0 blocks . size 12{""}

Calculating a resultant vector

If the perpendicular components A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} of a vector A size 12{A} {} are known, then A size 12{A} {} can also be found analytically. To find the magnitude A size 12{A} {} and direction θ size 12{θ} {} of a vector from its perpendicular components A x size 12{A rSub { size 8{x} } } {} and A y size 12{A rSub { size 8{y} } } {} , we use the following relationships:

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Source:  OpenStax, 2d kinematics. OpenStax CNX. Sep 04, 2015 Download for free at http://legacy.cnx.org/content/col11879/1.3
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