1.3 | Vectors and Scalars

What is a scalar, and what is a vector?

Units can be either vector, or scalar quantities. All that this refers to is whether a value has a direction or not. For example, distance and displacement, although similar, are different things, as the former is a scalar, whilst the latter is a vector.

Though not required, when vector quantities are denoted by a variable, the variable tends to have an arrow above it. For example, instead of \(v\), one would write \(\vec{v}\).

Combinations/resolutions of vectors

The skill of determining resultant vectors is extremely important, especially in topic 2; when an object is acted upon by multiple different unbalanced forces (which have different directions; vectors), which direction will the object actually move in? This is determined by the resultant vector— the combination of all vectors.

The method of vector resolution is just basic trigonometry. First, vectors should be moved such that their arrowheads touch the tails of the other vectors. This is referred to as the head-to-tail method.

For example, given 2 different forces with vectors \(\vec{a}\) and \(\vec{b}\), the head-to-tail method will make it much easier to see what the vector of the resultant force (denoted \(\vec{c}\)) is:

For a more complicated example; two vectors with magnitudes 75.0 and 53.0 (units unspecified) are given, where the vector with magnitude 53.0 is at an angle 34° from the horizontal. To find the resultant vector, the first step is always to use the head-to-tail method.

From here, the resultant vector \(\vec{x}\) can be found. However, it is still not completely straightforward. First, the vector with magnitude 53.0 must be resolved. 

Resolving vectors is the opposite of combining them; if a 2D vector has an awkward angle (like 34°), it must be converted into both horizontal and vertical components, so that it is easier to work with. In such cases, trigonometry can be used (the formulas for these are in the data booklet at 1.3, so you don't have to think about it too much).

In this case, the horizontal component can be denoted by \(\vec{y}\), and the vertical by \(\vec{z}\):

Now, since both the vector \(\vec{z}\) and the vector with magnitude 75.0 are on the same plane, they can be added. However, since they are vectors in different directions, one of them must be negative. It doesn’t really matter which one becomes negative, as long as opposite directions do not have the same sign (+ or -).

Subtracting the opposite vectors in the same plane leads to a simple triangle:

where the resultant vector \(\vec{x}\) can be easily found using the pythagorean theorem, since the vectors are aligned such that a right-angled triangle is formed, with \(\vec{x}\) as the hypotenuse:

\[x=\sqrt{{(43.9)}^{2}+{45.4}^{2}}=\boxed{63.2}\]