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# AS Physics: Scalar & Vector Quantities

A scalar has magnitude only. A vector has magnitude and direction.

Remember that S.I. units must be included for all quantities to define a magnitude.

Example scalars: distance, speed, work done, energy, power, time, mass

Example vectors: displacement, velocity, acceleration, momentum, force, impulse

## Vector Rules

Vectors are represented by arrows, the lengths of which indicate relative magnitudes. For example, the vector below has a size of 3 cm. Its direction is clearly indicated as well: The sum of vectors A and B (below) is R (the resultant):
A + B = R Drawn to scale, we simply join arrows one after the other. The resultant is the vector represented by the arrow that goes from the starting point to the end point.

## Subtracting

In vector terms, there isn't really such a thing as "subtracting" one vector from another. It's easier to think of it as adding a negative vector (i.e. the reverse of the vector being subtracted). e.g. A - B = R can be written: A + -B = R.
-B is the opposite vector to B ## Perpendicular Vectors

The resultant, R, of two perpendicular vectors, X and Y, is given by Pythagoras‘s Theorem: R² = X² + Y² ## Resolving Vectors

Perpendicular components of a vector are two vectors at right angles to each other that add up to give the original vector. This can be very useful indeed!

Remember your basic trigonometry:

sin q = opposite/hypotenuse
cos q = adjacent/hypotenuse
tan q = opposite/adjacent

[Click for a trigonometry refresher!]

For example, if X and Y are the horizontal and vertical components of R, we can find X and Y if we know another angle, say q as shown on the diagram: Note that the angle between X and Y must be set at 90° so that the components are perpendicular.

In this case: opposite represents X, adjacent represents Y and the hypotenuse represents R

From the sine and cosine relations:

vertical component, Y = Rcosq (since cosq = A/H = Y/R)
horizontal component, X = Rsinq

Perpendicular components of vectors act independently. For example a horizontal force will cause horizontal not vertical acceleration. Acceleration due to gravity downwards will only change vertical velocity, not horizontal velocity.