Physics and Physical Measurement Topic 1.3 Scalars and Vectors

Scalars Quantities b Scalars can be completely described by magnitude (size) b Scalars can be added algebraically b They are expressed as positive or negative numbers and a unit b examples include :- mass, electric charge, distance, speed, energy

Vector Quantities b Vectors need both a magnitude and a direction to describe them (also a point of application) b They need to be added, subtracted and multiplied in a special way b Examples :- velocity, weight, acceleration, displacement, momentum, force

Addition and Subtraction b The Resultant (Net) is the result vector that comes from adding or subtracting a multiple vectors b If vectors have the same or opposite directions the vector addition can be done simply: same direction : addsame direction : add opposite direction : subtractopposite direction : subtract

Co-planar vectors b The addition of co-planar vectors that do not have the same or opposite direction can be solved by using either… Scaled DrawingScaled Drawing –Vectors can be represented by a straight line segment with an arrow at the end Pythagoras’ theorem and trigonometryPythagoras’ theorem and trigonometry

Scaled Drawing method - triangle: b Choose an appropriate scale to fit the space b Draw one vector represented by straight arrow equal to the scaled length and with the arrow head pointing in the proper direction b Start drawing the second vector at the tip of the first b The resultant vector is the third side of a triangle and the arrow head points in the direction from the ‘free’ tail to the ‘free’ tip

Example a b+= R = a + b Measure the length and direction of the resultant in the drawing. Use your scale to convert to real life value.

Scaled Drawing - Parallelogram b Draw the two vectors tail to tail, to scale and with the correct directions b Then complete the parallelogram b The diagonal starting where the two tails meet and finishing where the two arrows meet becomes the resultant vector

Example a b+= R = a + b

More than 2 vectors b If there are more than 2 co-planar vectors to be added, place them all head to tail to form polygon b The resultant is drawn from the ‘free’ tail at the beginning to the ‘free’ tip at the end. b The order doesn’t matter!

Subtraction of Vectors b To subtract a vector, you reverse the direction of that vector to get the negative of it, then you simply add that vectors

Example a b- = R = a + (- b) -b

Multiplying Vectors b A vector multiplied by a scalar gives a vector with the same direction as the vector and magnitude equal to the product of the scalar and a vector magnitude b A vector divided by a scalar gives a vector with same direction as the vector and magnitude equal to the vector magnitude divided by the scalar b You don’t need to be able to multiply a vector by another vector

Resolving Vectors b The process of finding the Components of vectors is called Resolving vectors b Just as 2 vectors can be added to give a resultant, a single vector can be split into 2 components or parts

The Rule b Any vector can be split into two perpendicular components b These could be the vertical and horizontal components Vertical component Horizontal component

b Or these could be parallel to and perpendicular to an inclined plane

Doing the Trigonometry  Sin  = opp/hyp = y/  V  Cos  = adj/hyp = x/  V  V y x Therefore y =  V  sin  Which is the vertical component Therefore x =  V  cos  Which is the horizontal component  V cos  V sin 

Adding 2 or More Vectors by Components b First resolve into components (making sure that all are in the same 2 directions) b Then add the all the components in each of the 2 directions b Recombine them into a resultant vector using Pythagoras´ theorem