The dot product of two vectors: for A Level Further Maths

-

 

The dot product of two vectors: A Level further maths

The dot product or scaler product of two vectors, a and b, is defined as follows:

a.b = |a| |b| cos θ

Since |a|, |b| and cos t are scalers, the dot product is a scaler.

The dot product of unit vectors

i.i = |i| |i| cos 0 = 1 X 1 X 1 = 1
i.j = |i| |j| cos 90 = 1 X 1 X 0 = 0
j.j = |j| |j| cos 0 = 1 X 1 X 1 = 1

The dot product of two perpendicular vectors is zero.

E.g.
a = 3i + 4j   b = 5i - 12j  θ = 60
|a| = 5   |b| = 13

a.b = |a| |b| cos θ
a.b = 5 X13 X cos 60
a.b = 32.5

For more interactive simulations and worked examples, please use this link:




Comments

Post a Comment

Popular posts from this blog

Phase Difference between Two Points on a Wave and Path Difference Explained - interactive

-
Image
  In the above animation, five points on the wave are considered for the explanation.  The fully interactive applet is given below for you to practise. The phase of a point implies its direction of vibration on a wave. For example, both points A and E vibrate exactly the same way are said to be in phase: when A goes up so does B; the phase difference is either 0 0 or 360 0 . If you consider the motion of points A and C, on the other hand, when A goes down C goes up or vice versa. Therefore, A and C are said to be out of phase; the phase difference is 180 0 . That means the phase difference between two points on a wave - or two waves for that matter - can take any value between 0 0 and 360 0 .  The phase difference between points A and B, for instance, is 90 0 . From the above examples, it is clear there is a connection between the path difference between two points - or two waves - and phase difference. It is as follows: Phase difference = (path difference / wavelength) x 360 φ = x
Read more

Frequency Amplitude and Wavelength of a Transverse Wave - simulation

-
Image
Waves A wave carries energy from a point to another point in a medium / vacuum without the movement of the particles along with it. In the above animation, the wave moves from the left to the right; the particles, however, just vibrate vertically without moving along with the wave. Transverse Waves If the direction of the wave is perpendicular to the vibration of particles, it is called a transverse wave. E.g. Water waves, radio waves, microwaves Amplitude The maximum displacement of a particle in a wave is called amplitude. Intensity / loudness The intensity / loudness of a wave is proportional to the square of the amplitude. The greater the amplitude, the greater the intensity / loudness. Amplifier This is what an amplifier does: it increases the amplitude of an input signal, which in turn leads to the increased loudness or intensity. Frequency  - f The number of cycles of a wave that passes through a point in a unit time is called frequency. Wavelength  - λ The distance between
Read more