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

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Vectors[edit]

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:

ForceTriangle.jpg


Our point, P, is a plane travelling along the hypotenuse of this triangle at a speed of  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  4ms^{-1} along the horizontal and  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  {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 it's velocity would be  {4 \choose 3}ms^{-1} .

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

1. Draw a triagle represnting the vector.

2. Label all known values on traingle.

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\ = horizontal component^2 = 576. 576^\frac{1}{2} = 24. In i and js this is (24i+7j).