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.
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.