1.3.1 Distinguish between vector and scalar quantities, and give examples of each.
Scalars only have one dimension, size, while vectors have two: magnitude (size) and direction. Vectors are represented in print as bold and italicized symbols, for example: F.
Scalars | Vectors |
---|---|
Mass | Force |
Speed | Velocity |
Charge | Acceleration |
Distance | Displacement |
Energy | Momentum |
When a car moves backwards, its displacement is said to be negative. Any vector with a negative value indicates that the object is moving in the opposite direction. We usually use scalar values in normal conversation, such as the distance traveled by a car, because the direction does not matter. It is important to note that if a person has walked 10m forward and 10m back again, the distance traveled was 20m but their displacement is 0 because they are in the same position. Mathematically, this would be represented as 10m forward and -10m backward, which cancel out.
1.3.2 Determine the sum or difference of two vectors by a graphical method.
Parallel vectors
The sum of parallel vectors that run in the same direction can be determined by simple addition.
The sum of parallel vectors that run in the opposite direction can be determined by the subtraction of the smaller vector from the larger vector.
Vectors and scalars
Multiplying vectors by scalars functions like any ordinary equation. For example, F × 2 is 2F. It follows distributive properties -- 2(F + M) = 2F + 2M, and associative properties -- 2(MF) = 2F(M). When graphed, vectors multiplied by scalars become longer and vectors divided by scalars become shorter. Negative vectors go in the opposite direction of their positive counterparts. Note that when a vector is multiplied by 0, it is a null vector and has no magnitude or direction.
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Adjacent vectors
The sum of two vectors that are perpendicular or adjacent to each other can be determined through the Pythagoras theorem.
This can also be determined using basic trigonometry.
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1.3.3 Resolve vectors into perpendicular components along chosen axes.
Just as two vectors can be added to form a resultant vector, a resultant vector can be split into its two components. The basic rule of thumb is:
Consider the following example:
The two vectors that run diagonally can be split into their individual components for further calculation.
It should be taken into account that the plane of reference will not always be the paper. The object may be situated at a slope.
In this case, the force of the weight is not actually running straight downwards. We need to use the slope as a reference for the components of the vector.