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Vectors

What is a vector?

In our Physics course you can think of a vector as a measurement that requires 2 parts to fully record it. The two parts we will be dealing with are magnitude (an amount) and direction.

Vectors can be visually modeled as arrows with appropriate direction and lengths proportional to their magnitude.
Vectors are notated with an arrow above the symbol for that quantity (stop now and find at least 3 examples of these on the AP Physics Equation Table)

Here are 4 examples of vectors we will see throughout our course, what the parts of them are, and how they can be represented.

The other type of measurement we will be using is a scalar. Scalars only need a magnitude to be fully represented. Examples include time (5 sec), mass (8 kg), and height (2 m).

Why do they matter?

Vector quantities are often represented with an arrow at this level in Physics. The values can be added and subtracted through a simple graphical method or by following some mathematical routines. When you add vectors, though, the direction needs to be considered as well. For example, displacement is a measurement of how a final position differs from the original position. If a person moves 3 meters and then 4 meters how far away is she from where she started?

In this case you can see that the direction matters because she could be any distance between 1 meter away (ex. 1) to 7 meters away (ex. 2) to somewhere in between (5 meters away in example 3, but at different angles any value between 1 and 7 meters is possible).

Tip to tail addition of vectors in 1D

In this method for adding vectors you simply draw the first vector as an arrow and then

to add a vector, draw the second one from the end point of the first arrow (put the tail of the new vector at the tip of the 1st one).
to subtract a vector, flip the direction of the second vector and then draw the second one from the end of point of the first arrow.

The result of vector addition is called a resultant and is found by drawing a new arrow from the origin of the first vector to the tip of the last vector added (this works for adding more than just 2 vectors).

In the case of example A above, the resultant is a vector of magnitude 1m with a direction of -x (or to the left). Interestingly, it doesn't matter which order you draw these, as you can see at right, placing the 4 m left first, then coming 3 meters back to the right will produce the same resultant of 1 meter to the left.

These examples (variations on case A) are so simple that you really don't need to draw the vectors because they are both acting along the same axis. Anytime you need to add or subtract vectors on the same axis you can simply represent their direction as positive or negative. I just show them so you can see the ideas in a simple situation.

Tip to tail addition of vectors in 2D

When you have vectors act on more than one axis, you will likely need to use the geometry to find the magnitude and direction of the resultant. In our course we will most often see perpendicular vectors, which means we can use Pythagorean theorem and trig functions.

For example, C above, you find the magnitude of the resultant by using the a^2+b^2=c^2 with this is a right triangle to find the resultant is 5 m. Then use the trig function that relates the angle to the opposite and adjacent sides, which is tangent (θ) = o/a to find an angle of 60.6 degrees. The resultant vector can be described as 5 m at an angle of 60.6º above the X axis.

Finding components of vectors

There are times when we have an angled vector, most often an angled velocity or an angled force, that needs to be added to other vectors that are not on the same axis as it. It is MUCH easier to add vectors that are on the same axis as each other or perpendicular to each other, and we can make that happen if we split this vector into its vector components. What this means, is split up the angled vector into two simpler vectors along the x and y axes to see how it affects each axis separately.

To do this, we will make a right triangle where the hypotenuse is the vector we started with. Then identify the opposite and adjacent sides of the triangle we formed and use the appropriate trig functions. For example, a ball launched at a velocity of 60 m/s at an angle of 30 degrees above the horizontal can be split using this method:

Practice

Practice adding these vectors and checking your work here: At the physicsclassroom.com

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