1.3 Vectors and Scalars
Representation of a vector
A vector is represented by a line with an arrow at its end or at its middle, as shown by the two equal vectors a to the right.
The arrow indicates the direction.
The length indicates the magnitude.
Components of a vector
A vector may be decomposed into one vertical and one horizontal component, as follows:
A vector’s magnitude may be found from its vertical and horizontal components, through the formula (known as Pythagoras Theorem): A^2 = Ax^2 = Ay^2 .
A vector’s direction (angle with the horizontal) may be calculated by means of the following formula: θ = tan^-1 Ay/Ax
When two or more vectors are added or subtracted, a resultant vector is formed.
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
Note that some quantities appear to be the same, such as velocity and speed, both representing distance over time, the difference is that velocity has a direction whilst speed does not.
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 1.3.1 – Resultant force of two opposing vectors
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 1.3.2 – Resultant force of two concurrent 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 1.3.3 – Graphical method of solving adjacent vectors
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
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)
Scalar multiplication is demonstrated in figure 1.3.5 below:
Figure 1.3.5 – Scalar multiplication and division of vectors
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 1.3.6 and 1.3.7:
Figure 1.3.6 shows a diagram of the forces acting on a block being pushed along a smooth surface
Figure 1.3.6 – Forces acting on a block
Figure 1.3.7 shows the same diagram but with the surface and pushing forces broken down into their components:
Figure 1.3.7 – Forces acting on a block broken into their components
Sometimes the plane of reference will not be parallel to the page, such and example is shown in figure 1.3.8 below:
Figure 1.3.8 – Component forces of a block on a slope
SCALARS AND VECTORS
- it must have magnitude.
- it must have direction.
- it must satisfy parallelogram law of vector addition.
TYPES OF VECTORS
- The sum of a finite vector and the zero vector is equal to the finite vector
- The multiplication of a zero vector by a finite number n is equal to the zero vector
- The multiplication of a finite by a zero is equal to zero vector
LAWS OF VECTOR ALGEBRA
- (Commutative law of addition)
- (Associative law of addition)
ADDITION OF VECTORS
TRIANGLE LAW OF VECTOR ADDITION
- || is maximum, if cosθ = 1, θ = 0° (parallel vector)
- || is minimum, if cosθ = –1, θ = 180° (opposite vector)
- If the vectors A and B are orthogonal,
PARALLELOGRAM LAW OF VECTOR ADDITION
POLYGON LAW OF VECTOR ADDITION
- Resultant of two unequal vectors cannot be zero.
- Resultant of three coplanar vectors may or may not be zero.
- Minimum no. of coplanar vectors for zero resultant is 2 (for equal magnitude) and 3 (for unequal magnitude).
- Resultant of three non coplanar vectors cannot be zero. Minimum number of non coplanar vectors whose sum can be zero is four.
- Polygon law should be used only for diagram purpose for calculation of resultant vector (For addition of more than 2 vectors) we use components of vector.
- If , then is a null vector.
- Null vector or zero vector is defined as a vector whose magnitude is zero and direction indeterminate. Null vector differs from ordinary zero in the sense that ordinary zero is not associated with direction.
- is called a unit vector. It is unit less and dimensionless vector. Its magnitude is 1. It represents direction only.
- If , then and , where are unit vectors of A and B respectively.
- A vector can be divided or multiplied by a scalar.
- Vectors of the same kind can only be added or subtracted. It is not possible to add or subtract the vectors of different kind. This rule is also valid for scalars.
- Vectors of same as well as different kinds can be multiplied.
- A vector can have any number of components. But it can have only three rectangular components in space and two rectangular components in a plane. Rectangular components are mutually perpendicular.
- The minimum number of unequal non-coplanar whose vector sum is zero is 4.
- makes 45° with both X and Y-axes. It makes angle 90° with Z-axis.
- makes angle 54.74° with each of the X, Y and Z-axes.
- If then angle between and is .
- Magnitude of a vector is independent of coordinate axes system.
- Component of a vector perpendicular to itself is zero.
- Resultant of two vectors is maximum when θ = 0°, Rmax = A + B
- Resultant of two vectors is minimum when θ = 180° , Rmin = A – B
- The magnitude of resultant of and can vary between (A + B) and (A – B)
SUBTRACTION OF VECTORS
RESOLUTION OF A VECTOR
RECTANGULAR COMPONENTS OF A VECTOR IN PLANE
RECTANGULAR COMPONENTS OF A VECTOR IN 3D
Do not resolve the vector at its head.
PRODUCT OF TWO VECTORS
SCALAR OR DOT PRODUCT
PROPERTIES OF SCALAR OR DOT PRODUCT
- = A (B cosθ) = B (A cosθ)
- Dot product of two vectors is commutative.
- Dot product is distributive.
- = (Ax Bx + Ay By + Az Bz)
VECTOR OR CROSS PRODUCT
PROPERTIES OF VECTOR OR CROSS PRODUCT
- (not commutative)
- (follows distributive law)
- The cross product of two vectors represents the area of the parallelogram formed by them.
- A unit vector which is perpendicular to A as well as B is
- tan θ =
- If , then
- If then angle between and is .
- If then
- Division by a vector is not defined. Because, it is not possible to divide by a direction.
- The sum and product of vectors is independent of coordinate axes system.
CONDITION OF ZERO RESULTANT VECTOR
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.
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.