Posts Tagged ‘Maths’

Vector basics

I wanted to get back to explaining Maxwell’s equations, which I mentioned in this blog of the statue to James Clerk Maxwell that is in Edinburgh. Before I do that I thought I would cover some very basic mathematics, namely the basic idea of vectors.

In physics and mechanics, a vector is something which has both size (magnitude) and direction. If the quantity has only a size, we call it a scalar. So, for example, in physics we differentiate between speed and velocity, even though in everyday language they are used interchangeably.

If we say a car is moving at 50 km/h, that is a speed, and hence is a scalar. But, if we were to say it is moving at 50 km/h due North, that is a velocity, and hence a vector as we have given it both a size and direction.

Vectors are very important in physics, and one of the most important and widely used vectors is force. We measure force in Newtons (named after Sir Isaac Newton), but a force has both a size and a direction. We usually denote a vector by either putting a line or arrow above the symbol, or by using a bold font. I will use an arrow above the symbol, so for example v would be a scalar, but \vec{v} would be a vector.

Any vector in 3-dimensional space can be split into 3-components. This is often useful, as unlike scalars which add simply, when we add vectors we need to take into account their directions. As an example, suppose we have two forces \vec{F_{1}} which has a size of 7N in the x-direction, and \vec{F_{2}} which has a size of 5N at 25^{\circ} to the x-axis. The combined force is not 12N, as the two forces are not acting in exactly the same direction.

A 7N and a 5N force at 25 degrees to each other, with the 7N force acting along the x-direction.

A 7N and a 5N force at 25 degrees to each other, with the 7N force acting along the x-direction.

To find the combined (resultant) force, we need to split the two forces up into their x and y-components. To do this we simply use trigonometry, and note the unit vectors \hat{x} and \hat{y}, which are vectors with a size of unity (1) in the x and y-directions respectively. We can then split each of the forces \vec{F_{1}} and \vec{F_{2}} into their respective x and y-components.

\vec{F_{1}} = 7\cos(0) \hat{x} + 7\sin(0) \hat{y} = 7\hat{x} + 0\hat{y}

\vec{F_{2}} = 5\cos(25) \hat{x} + 5\sin(25) \hat{y} = (5 \times 0.9063)\hat{x} + (5 \times 0.4226)\hat{y} = 4.53 \hat{x} + 2.11 \hat{y}

From this we can write that the total force in the x-direction is 7 + 4.53 = 11.53 \hat{x} and the total force in the y-direction is 0 + 2.11 = 2.11 \hat{y}. To then find the resultant force, we need to combine these two components as follows

The size of the resultant force R can be calculated using Pythagoras' theorem, its direction using trigonometry.

The size of the resultant force R can be calculated using Pythagoras’ theorem, its direction using trigonometry.

To calculate the size of the resultant vector we just use R^{2} = x^{2} + y^{2} so here R^{2} = (11.53)^{2} + 2.11^{2} = 157.0 + 4.45 = 137.39 so R=\sqrt{137.39}=11.72 N. To calculate the angle \theta we note that \theta = \arctan \left( \frac {2.11} {11.72} \right) = 10.2^{\circ}.

So, as we can see, the resultant of these two forces is a force of size 11.72N which is at an angle of 10.2^{\circ} to the x-axis.

In a series of future blogs I will go on from these basic ideas of vectors to talk about vector fields, which we need to understand in order to understand Maxwell’s equations.

A vector field

A vector field. The length of each arrow represents the strength of the field at that location, the direction is given by the orientation of the arrow.

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