Physics vectors and scalars

1. Vectors and scalars
IB 11th Grade Physics Lesson 2

2. 1.3.1 Distinguish between vector and scalar quantities,
and give examples of each.
• When expressing a quantity we give it a number and a unit (for example, 12 kg),
this expresses the magnitude of the quantity. Some quantities also have direction,
a quantity that has both a magnitude and direction is called a vector. On the other
hand, a quantity that has only a magnitude is called a scalar quantity. Vectors are
represented in print as bold and italicised characters (for example F). Below is a
table listing some vector and scalar quantities:
• Table 1.3.1 - Vector and scalar quantites
Scalars Vectors
Speed Velocity
Temperature Acceleration
Distance Displacement
Area Force
Entropy Momentum
Volume Drag

3. 1.3.2 Determine the sum or difference of two
vectors by a graphical method.
• The difference of two vectors
When adding vectors, we need to take both the magnitude and direction
into account. Often, we will have situations where two vectors have
opposite directions, in this case, we simply subtract the smallest
magnitude from the largest one. This is demonstrated in figure 1.3.1
below:
Figure 1.3.2 - Resultant force of two opposing vectors

4. The sum of two vectors
Sometimes we will have situations where two forces are acting in the same
direction. In the situations we simply add together the magnitudes of both
vectors. This is demonstrated in figure 1.3.2 below:
Figure 1.3.3 - Resultant force of two concurrent vectors

5. Adjacent vectors
In certain situations, we will need to work out the angle between two
adjacent vectors. In order to do this graphically we draw a scale diagram with
the tail of one vector at the head of the other, we then draw a line connecting
the other head and tail. To get the magnitude of the new vector, we simply
measure it. This is demonstrated in the diagram below:
Figure 1.3.4 - Graphical method of solving adjacent vectors

6. Alternatively, we can use trigonometry for a faster and more accurate result. This
is demonstrated in figure 1.3.4 below:
Figure 1.3.4 - Trigonometric method of solving adjacent vectors

7. Scalar multiplication
We can also multiply (and divide) vectors by scalars. When doing so we follow a set of
rules:
• Multiplying by 1 does not change a vector 1 v = v
• Multiplying by 0 gives the null vector 0 v = 0
• Multiplying by -1 gives the additive inverse -1 v = -v
• Left distributivity: (c + d)v = cv + dv
• Right distributivity: c(v + w) = cv + cw
• Associativity: (cd)v = c(dv)

8. Scalar multiplication is demonstrated in figure 1.3.5 below:
Figure 1.3.6 - Scalar multiplication and division of vectors

9. 1.3.3 Resolve vectors into perpendicular components along chosen axes.
When working with adjacent vectors that do not form a 90° angle, it is often useful to brake
certain vectors into component vectors so that they are concurrent with the other vectors. To
do this, we draw two vectors, one horizontal and the other vertical to our plane of reference.
We then use trigonometry to work out the magnitude of each new vector and figure out the
resulting force. This is shown in figure 1.3.7 and 1.3.8:
Figure 1.3.7 shows a diagram of the forces acting on a block being pushed along a smooth
surface:
Figure 1.3.7 - Forces acting on a block

10. Figure 1.3.8 shows the same diagram but with the surface and pushing
forces broken down into their components:
Figure 1.3.8- Forces acting on a block broken into their components

11. Sometimes the plane of reference will not be parallel to the page, such and
example is shown in figure 1.3.9 below:
Figure 1.3.8 - Component forces of a block on a slope