A-level Mathematics/OCR/M1/Force as a mk Vector

Vectors

A vector is a quantity that has both a magnitude (or a size) and a direction. The opposite of vectors are scalars. Scalars only have a magnitude. There is no direction. For example, speed is a scalar as speed is the same regardless of direction. This is best illustrated as a triangle:

Our point, P, is a plane travelling along the hypotenuse of this triangle at a speed of $5m{s}^{-1}$. Its velocity, however, is not 5. As velocity is a vector and has both magnitude and direction, the speed of P is equal to moving at a velocity of $4m{s}^{-1}$ along the horizontal and $3m{s}^{-1}$ along the vertical.

There are several different ways of writing this as a vector. One of the most common is the i and j notation. Where i is the horizontal component of the velocity and j is the vertical component of the velocity. Using this notation, our plane would have a velocity of (4i + 3j)

Another common way of writing vectors is in the form of $(\genfrac{}{}{0ex}{}{x}{y})$ where x is the horizontal component and y is the vertical component. Using our plane as the example, is this vector form its velocity would be $(\genfrac{}{}{0ex}{}{4}{3})m{s}^{-1}$.

To change a Vector into its horizontal and vertical components we:

1. Draw a triangle representing the vector.

2. Label all known values on triangle.

3. Use trigonometry to solve.

E.g. A force, P, with magnitude 25N has a direction of $arcsin$ $\frac{7}{25}$ (arcsin is the opposite of $sin$.), find the horizontal and vertical components of P

Triangle

Label triangle:

Use trigonometry: is $\arcsin \theta$ is $\frac{7}{25}$ then $\theta$ is $sin$ $\frac{7}{25}$. Sin is O/H. Therefore the vertical component of P is 7. The Horizontal component can be found by using Pythagoras' theorem or recognising 7, 24, 25 as a Pythagorean triple. Pythagoras' theorem says that $a^2$+$b^2$=$c^2$ where c is the hypotenuse and a and b are the adjacent and opposite (order does not matter). Therefore $25^2-7^2$ = $horizontalcomponen{t}^2$ = 576. $576^{\frac{1}{2}}$ = 24. In i and js this is (24i+7j).