A-level Physics/Forces and Motion/Scalars and vectors

Vectors and scalars are mathematical constructs which physicists employ. Some physical quantities are represented by scalars and some by vectors and corresponding operations are employed upon them while dealing with them. Vector quantities have a direction associated with them while scalars are treated like simple numbers.

The following are some examples of quantities that are represented as scalars and vectors.

Scalars

The following quantities have a magnitude, and perhaps a sign, but no direction associated with them, and are examples of scalars:

• distance
• speed
• time
• mass
• energy
• density
• temperature

Normally scalar quantities follow the basic algebraic rules for any mathematical manipulation such as addition, subraction, multiplication or division (as we do with numbers).

Addition of scalars

Adding scalars is simple, all you need to do is to add the numbers together. For example, 5m + 3m = 8m, or 76b + 23b = 99b

Multiplication and division of scalars

Multiplying and dividing scalars is the same as multiplying and dividing normal numbers.

You should also remember to multiply and divide the units, so that you can check your answers are given in the correct units. For example, if you were finding the area of a surface: $3m\times 3m=9m^{2}$. The unit of area is $m^{2}$, so this is correct.

Vectors

The concept of direction establishes a relationship between two points in space; that is, the "direction" from one point to another. For example, the direction from point A to point B could be designated A-to-B while the opposite direction would be in that case B-to-A. Direction is dimensionless; that is, it has no measurement units and represents only a line designating the sense of from-to (from A to B) with no sense of "how much" which is considered the "magnitude" of a measurable quantity.

"Magnitude" provides a sense of "how much" (or "how many") of a measurable quantity. The term five miles has a magnitude of five units of measure; this unit of measure is miles.

When magnitude ("five" miles) is coupled with direction (let's say north; which is the dimensionless direction from me to the North Star) we obtain "five miles north"; this is a "vector". A "vector" has both magnitude and direction. A special type of vector has a magnitude of one in a given direction and is called a “unit vector” for that direction. A quantity that has only magnitude but has no associated direction is a "scalar", as described earlier.

Two vectors that have the same direction and magnitude are equal; a vector from me that is "five miles north" is equal to a vector from you that is "five miles north"; wherever you are. However, the position obtained by moving from me to "five miles north" of me is not a vector. The vector is the "displacement" that consists of a magnitude of distance (five miles) and a direction (north). A position can be represented by a beginning reference point (you; wherever you are) and a vector (five miles north), but a vector alone is not a location; it must have a reference location to be meaningful as a position.

The following quantities have both a magnitude and a direction associated with them, and therefore are vectors:

• Displacement (e.g., five miles north)
• Velocity (50 metres per second, bearing 60 degrees 15 seconds)
• Acceleration (32 feet per second per second straight up)
• Weight (your mass straight down)
• Force (the amount of energy in a given direction)

Vectors can be represented in any set of spatial dimensions, though typically they are expressed in 2-D or 3-D space.

Multiplication

Multiplying Vectors to Scalars

When you multiply a vector by a scalar, the result is a vector. Its direction is unchanged if multiplied by a positive scalar and its direction is reversed when multiplied by a negative scalar. The vector's magnitude is simply multiplied by the scalar.

Multiplying Vectors to Vectors

There are two different kinds of multiplication when you multiply two vectors together. There is the dot product, and there is the cross product. The multiplication of two vectors is outside of the scope of an A-level physics course, but you can find out about them on Wikipedia.

Addition of vectors

Vectors can be added like scalars as long as they are facing exactly the same direction. If the vectors are in opposite directions, you must subtract one from the other, and unless stated otherwise, you should use the common conventions that:

• up is positive, and down is negative, and
• right is positive and left is negative.

When the vectors aren't in a straight line, you must use another method to find their sum.

Pythagoras' theorem

If the two vectors are perpendicular to each other, it is possible to find the total vector using Pythagoras' theorem, with the resultant vector being the hypotenuse of the right-angled triangle.

The direction of the resultant can be found using the formula: $\tan \phi ={\frac {b}{a}}$, where $a$ is one of the sides touching (adjacent to) the angle, $b$ is the side opposite the angle, and $\phi$ is the angle of the resultant vector.

Notice that vector a has no effect in the direction of vector b, and similarly, vector b has no effect in the direction of vector a. When two vectors are perpendicular to each other, it is said that they act independently of each other.

Resolving vectors into two perpendicular components

A vector can be broken down into components, which are perpendicular to each other, so that the vector sum of these two components, is equal to the original vector. (Usually, it is interesting to break down a vector into two perpendicular components, such that one is vertical and the other horizontal. However, the components do not have to be chosen to be vertical and horizontal always; they only need to be perpendicular to each other). Splitting a vector into two components is called resolving the vector. It is the reverse of using Pythagoras' theorem to add two perpendicular vectors, and so adding the two components will give you the original vector. There are many uses for vectors that have been split in this way.

Resolving a vector requires some simple trigonometry. In the diagram, the vector to be resolved is the force, $F$. For angle $A$:

• the horizontal component of $F$: $F_{x}=F\ \cos \ A$, and
• the vertical component of $F$: $F_{y}=F\ \sin \ A$.

Note that the two components do not have to be horizontal and vertical. The angle $A$ can be changed to any required direction, and both components will still be perpendicular to each other.