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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:

3 cm long vector, pointing right

Adding

The sum of vectors A and B (below) is R (the resultant):
A + B = R

resultant of two vectors

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).

Vector B and vector -B

e.g. A - B = R can be written: A + -B = R.
-B is the opposite vector to B

subtraction of two vectors: simply add the negative value of the subtraction!

Perpendicular Vectors

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

resultant of two perpendicular vectors, X and Y

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:

resultant of two perpendicular vectors, X and Y

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.

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