8.01x - Module 03.01 - What is a vector, what is a scalar.

Transcript

what is the difference between a scaler and a vector there a huge difference a scaler is uniquely determined by one number temperature is a scaler one number is all it takes Mass your weight that's a scaler power energy those are scalers the speed of your car is a scaler one number tells it all the size of my shoes which in the United States I think is 11 to 12 and in Europe 44 to 45 but in all these cases one number tells the entire story that is not the case with vectors vectors have a magnitude and they have a Direction so you cannot characterize a vector by simply one number you need more than one number examples of vectors that we will see in this course are for instance velocities acceleration force momentum angular momentum torque impulse and position Factor the position of an object in three-dimensional space cannot be characterized with one number so a vector needs more than one number to specify it it has a magnitude and it has a Direction let us examine a vector a and I will write an arrow over the vector to indicate that this an arrow over the a to indicate that it is a vector some books write an a in bold which then means it's a vector I will not do that because I can't write bold so I will always use the notation a with a arrow over it and let's examine a vector in three dimensions for instance threedimensional space let's make a drawing x y z and we have a vector a like so this is a and I will project this a onto the XY plane and then I will further project it onto the y axis we call this a of Y I project it onto the xaxis which is a of X and I will project it onto the z-axis which is a of Z we call this a cartisian coordinate system three numbers uniquely Define this Vector the numbers ax a y and a z uniquely Define this vector and the vector notation you may see in certain books are as follows a or some books write you an A in bold and then they may simply give you these three components a of x a of Y and a of Z I will not do that I prefer a different notation I will write almost always that a with an error over it is the X component of a in the direction of X this is the unit Vector in the plus X Direction plus a of Y unit Vector in the y direction plus a of Z unit Vector in the Z Direction so what I have here I have literally written down this Vector a in ter terms of the sum of three vectors X roof is the unit Vector in the X Direction y roof is the unit Vector in the y direction and Z roof is the unit Vector in the plus Z Direction and so this is a vector in the X direction of course if a of X is negative then it would be in the negative X Direction and this is a vector in the y direction and this is a vector in the Z Direction and the sum of these three vectors make up a some books prefer instead of the X roof I roof for x j roof for y and K roof for Z I will not use that but you will see it very often and some books may even prefer a bolt I a bolt J and a bolt K so you will see all kinds of possibilities this is the one that I will prefer what now is the length of this Vector which we often refer to as the magnitude of that Vector for that we have various notations some book simply write an a not in bold some books write the vector notation with two bars on either side indic it is the magnitude of that vector and some books simply write an a with two bars on either side I may use either one of these three as it suits me in any case the magnitude of that Vector call it the length of that Vector equals the square root of a X2 + a y^ 2 plus a z^ 2 this Itself by way is a scaler it's just one number I'll give you an example let's have a velocity vector and the velocity Vector is given by 3 x roof plus 2 y roof minus 4 Z roof what this means is that if we assume that distances are always measured in meters and times are measured in seconds it means that the component in the X direction is + 3 m/s the component of the Velocity in the y direction is also plus and it is in this case plus two the component of the Velocity in the Z direction is 4 m/s but it is in the minus Z Direction the magnitude of this velocity Vector equals the square root of 3 squ which is 9 + 2 square which is 4+ minus 4 SAR which is 16 that's the square root of 29 m/ second this is the speed and so this is a scalar