Vectors - LeFonch

What are Vectors?

In physics there are both scalar and vector quantities.
Scalar quantities consist of only a magnitude and are independent of direction.
Think of for example mass and time. An object can't have a mass in any sort of direction, and thus it is scalar.

Scalar quantities can only be positive, as they are not based on direction.
Other types of units need to have a direction to be fully defined.
These types of units are called vectors.
Vectors have both magnitude and direction.
This means that the unit is directed towards a certain point. For example an object could be moving with a velocity of 15 m/s to the east.

Vector quantities can be either positive or negative, depending on their direction.
In a chosen direction, a vector quantity is positive, and in the opposite direction of the chosen one it is negative.

Vectors are drawn as arrows.
The length of the arrow is its magnitude, and the direction the head points is the direction the unit is pointed.
Magnitude is always proportional. If you were to double the length of the arrow, you would have doubled

A vector can be multiplied by a scalar in order to change its magnitude.
Multiplying vectors changes the length of the vector.
Values larger than one increase the length of the vector and values less than one will decrease the length of the vector.
Multiplying a vector by a scalar with a value of 0 makes the vector not exist. Imagine for example, how much distance a person could walk in 0 seconds?

Vectors can also be added or subtracted from one another as long as they represent the same physical quantity, depending on their direction.
If two vectors of the same quantity face the same direction, meaning they are parallel, then their magnitudes can directly be added to one another.
If the vectors face the opposite direction however, their magnitudes are subtracted from one another instead.
Remember to always keep vectors proportional to one another. If vector B has double the magnitude of vector A, it should be exactly twice as long.

If the vectors aren't aligned then different methods can be used to add them together.
For example in a situation where at least some angles are known, trigonometry can be used to align the vectors and thus add them together.
However, if you are not given such information then you can simply draw the vectors instead.
Vectors can be moved around freely as long as their magnitude and direction stays the same.
Thus you can bring two vectors end to head to tail and add them together by drawing a line that stretches from the tail of the first vector and to the head of the last vector.
If drawing the line forms a right triangle, then the Pythagorean theorem can be used to find the magnitude of the sum vector.
If the head of the last vector is at the tail of the first vector, then the value cancels itself out and it is zero.
For example, in the image below, vector v can be moved from its original position to the head of vector u. Then they can be added together by drawing a line from the tail of u to the head of v.

Solving vectors usually requires them to be split into their components.
Trigonometry can be used to find the values of a vector along the x and y axis for example, which can then be added or subtracted from other vector components to find the sum of two vectors.

Whether a quantity is a vector or scalar can be determined based on the units it is composed of.
As vectors can have negative signs to indicate direction, the way they are determined is similar to how the signs of products of positive and negative numbers are determined.

If a vector quantity is multiplied or divided by a vector, then any negative sign would cancel out, and the indication of direction would be lost.
Thus a vector multiplied or divided by a vector must be scalar.

If a vector quantity is multiplied or divided by a scalar, the sign does not change as scalar quantities are always positive.
This means the quantity can still indicate direction, and thus the resulting quantity is vector as well.

If a scalar is multiplied or divided by a scalar, then the final quantity is still scalar, as neither quantities could show direction to begin with.

Sources

Kognity - Physics Book