Scalar and vector quantities

In a nutshell

Physical quantities fall under two different categories, called scalar quantities and vector quantities. Scalar quantities have a magnitude but no direction, while vector quantities have both magnitude and direction. Vectors parallel to each other can be added and subtracted directly, but vectors perpendicular to each other must have Pythagoras' theorem applied first.

Definitions

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Scalar quantities

Scalar quantities are quantities that have a magnitude, described by a number, but have no direction. Scalars and and subtract in the same way regular numbers do, but they always must have the same units when they are added or subtracted.

Scalar quantities also multiply and divide in the same way as regular numbers, though the units do not need to be the same. The units of the final result however must be calculated correctly.​​

Example

Time is a scalar quantity, as there is no direction associated with time. 

Adding $$20\,s$$ to $$30\,s$$ is done in the usual way of adding numbers together.

$$20+30 = 50\,s$$

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Vector quantities

Vector quantities are quantities that have both magnitude and direction. Unlike scalars, vectors can only be directly added and subtracted if they are pointing in the same direction. If they aren't, more complicated processes are necessary to find a sum, or resultant, of vector quantities.

Differences between scalar and vector quantities

Some scalar quantities have the same units as vector quantities. For example, the displacement of an object is a vector that points from the object's start position to its end position. It is independent of the path that the object took to get there. 

In contrast, the distance the object travelled does depend on the path it travels.

Note: Only vectors have negative values. A negative vector indicates that it points in the opposite direction to the positive.

Example

Speed and velocity are two quantities with the same units. Which is a scalar and which is a vector?

Speed is a scalar, as it only describes how fast an object is moving. Velocity is a vector as it also describes the direction the object is moving. A negative velocity would mean that an object is travelling backwards with a certain speed.

Adding and subtracting vectors

Parallel vectors

Vectors that point in the same line can be added and subtracted as normal. One direction is defined as positive, while the other is negative, so that any vector that points opposite to the positive direction is given a negative sign. Then, the vectors are added as normal to find the resultant vector.

Example

Calculate the resultant of the following two forces.

Define one direction to be positive - in this case, the $$7N$$ force points in the positive direction, while the $$4N$$ force points in the negative direction.

Next, subtract the two forces.

$$7 - 4 = 3N$$​​

 The resultant of the two forces is $$\underline{3N}$$.

Perpendicular vectors

Perpendicular vectors are vectors that act at right angles to each other. They cannot be directly added or subtracted from each other.

In order to find the resultant of two perpendicular vectors, they should be represented as arrows, with the first vector ending where the second vector starts. Then the resultant is drawn starting from the start of the first vector, pointing to the end of the second, forming a vector triangle. The magnitudes of the vectors are represented by the sizes of the lines.

The resultant force can then be calculated by using Pythagoras' theorem. 

The direction of the resultant force can also be calculated by using trigonometry, and is sometimes called a bearing. A bearing is the clockwise angle that a vector makes with the North direction.

Example

Two forces, one of magnitude $$5N$$ and one of magnitude $$12N$$, act perpendicular to each other. Calculate the resultant force's magnitude and direction from the horizontal.

First, draw out the vector diagram.

Next, use Pythagoras' theorem to calculate the resultant force.

$$R = \sqrt{5^2 + 12^2}$$​​

​$$R = 13N$$​​

Next, use trigonometry to find the force's direction.

$$\tan\theta = \dfrac{opp}{adj}$$​​

​$$\tan\theta = \dfrac{12}{5}$$​​

​$$\theta = 67.4\degree$$​​

The resultant force has a magnitude of $$\underline{13N}$$, with a direction of $$\underline{67.3\degree}$$ from the horizontal.