how to add vector components

College Physics Supplementary Handout 1: Review Vectors and Coulomb Force

Adding Vectors:

Vectors, as oppose to such scalar quantities as time, mass, charge, etc. that do not have direction but have just magnitude, are quantities that have direction as well as magnitude (size, strength).� When adding or subtracting and� multiplying or dividing vectors, we use the scalar algebra for the components of the vector.� Another words, adding two vectors is not like adding two scalars.� Here is an example that contrasts adding two quantities .

Adding vectors� (Geometric method):

To add two vectors graphically, for example, , bring B�s tail to A head without changing the direction of B.� Draw an arrow from the tail of A to the tip (or head) of B.� This is Head to Tail or toe� method!� You cam also use parallelogram method to obtaion the same result.� Both methods are shown below.

Adding Vectors (algebraic Methods):

Add two distances:� A movement of 3 mi north than 4 miles east.� The resultant total distance (adding two distances) is 7 mi.� Here distance is a scalar.� Therefore, we used the scalar (one that you learned in High School) algebra to add them

Add two vectors:� A displacement of 3 mi north, then a second displacement of 4 mi east.� The resultant displacement is not 7 mi!� Instead, we must consider the directions as well the magnitudes.

In the example above left, the resultant vector C is obtained by adding two vectors A and B.� The arrows above each symbol indicate that these quantities have directions assigned to them.�� Note that the resultant is in the general direction of NE and 5 miles ( ).� When we need to decompose a vector into its components that is into two vectors mutually perpendicular to each other we need to undo what we did in the above example.� To do this we need a little trigonometry, namely sine, cosine, and cotangent.

Here are the definitions of sine and cosine.

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We can apply this to a vector.� Suppose we have a force of 20 N in the direction of 30o north of east.�� Let�s decompose this vector into its Cartesian components (an x direction component and a y direction component).

Therefore, a 20 N force is equivalent to a 17.3 N force in the north direction (y-direction, and a 10 N force in the east direction (x-direction).

What if the force direction were opposite, that is SW?� Let�s work out this case, keeping in mind that vector components in the south( negative y direction) and west (negative x �direction) should be fitted with a negative sign in front after using the process above.

Note that the calculations are the same,

except the signs,

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In mathematical language this is written as

where the sing above x and y are hats,

denote x and y directions.

When we add several vectors, these signs

will be important (see the upcoming example).

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An application of Coulomb�s law:

Three charges 1 C, 2 C and -3 C are situated at the corners of a right triangle as shown.� Find the net force on 1 C charge.

We first need to compute the magnitude of the forces F₂, and F_-3.�� Distance needed for F_-3 can be obtained from definition of sine,

opposite side = 10xSin (30) =10 (0.5) = 5 m

The total force is not 1.8x10⁺⁸+10.8x10⁺⁸ N!!!Because these forces are not in the same direction.� We will find the total force in class! See your notes.

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