What are scalar and vector quantities?

Scalar quantities have only magnitude, while vector quantities have both magnitude and direction.

Scalar quantities are physical quantities that have only magnitude, such as mass, temperature, and time. They can be added, subtracted, multiplied, and divided using the usual arithmetic operations. For example, if a car travels 100 km in 2 hours, its average speed can be calculated as 50 km/h by dividing the distance travelled by the time taken.

Vector quantities are physical quantities that have both magnitude and direction, such as displacement, velocity, and force. They cannot be added, subtracted, multiplied, or divided using the usual arithmetic operations. Instead, they are combined using vector addition, subtraction, and multiplication. For example, if a car travels 100 km north and then 50 km east, its displacement can be calculated as the vector sum of the two displacements, which is approximately 111 km at an angle of 26.6 degrees east of north.

In mathematics, vectors are often represented using column matrices or boldface letters, such as $\mathbf{a}=\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}$, where $a_1$, $a_2$, and $a_3$ are the components of the vector in three-dimensional space. The magnitude of a vector $\mathbf{a}$ is denoted by $|\mathbf{a}|$ and is given by $|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}$. The direction of a vector can be specified using angles or unit vectors, which are vectors of magnitude 1 that point in the same direction as the original vector.