# 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 ${\displaystyle 5ms^{-1}}$ . 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 ${\displaystyle 4ms^{-1}}$  along the horizontal and ${\displaystyle 3ms^{-1}}$  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 ${\displaystyle {x \choose y}}$  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 ${\displaystyle {4 \choose 3}ms^{-1}}$ .

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 ${\displaystyle arcsin}$  ${\displaystyle {\frac {7}{25}}}$  (arcsin is the opposite of ${\displaystyle sin}$ .), find the horizontal and vertical components of P

Triangle

Label triangle:

Use trigonometry: is ${\displaystyle \arcsin \theta }$  is ${\displaystyle {\frac {7}{25}}}$  then ${\displaystyle \theta }$  is ${\displaystyle sin}$  ${\displaystyle {\frac {7}{25}}}$ . 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 ${\displaystyle a^{2}}$ +${\displaystyle b^{2}}$ =${\displaystyle c^{2}}$  where c is the hypotenuse and a and b are the adjacent and opposite (order does not matter). Therefore ${\displaystyle \ 25^{2}-\ 7^{2}\ }$  = ${\displaystyle horizontalcomponent^{2}}$  = 576. ${\displaystyle 576^{\frac {1}{2}}}$  = 24. In i and js this is (24i+7j).