Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) PDF Download

Vectors - Introduction

  • There are physical quantities like force, velocity, acceleration and others that are not fully determined by their numerical data. 
  • For example, a numerical value of speed of motion, or electric or magnetic field strength, not give us the information about direction it move or direction they act.
  • Such quantities, which are completely specified by a magnitude and a direction, are called vectors or vector quantities and are represented by directed line segment.
  • Thus, a vector is denoted as Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)where the point A is called the tail or start and point B, the head or tip.
  • The length or magnitude or norm of the vector a or Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) is 

Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

Therefore, the length of the arrow represents the vector's magnitude, while the direction in which the arrow points, represents the vector's direction.

Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

A vector with no magnitude, i.e., if the tail and the head coincide, is called the zero or null vector denoted Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

Collinear, Opposite and Coplanar Vectors

Two vectors are said to be equal if they have the same magnitude and direction or if by parallel shift or translation one could be brought into coincidence with the other, tail to tail and head to head.

Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)Vectors are said to be collinear if they lie on the same line or on parallel lines.
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

  • Vectors,Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)in the above figure are collinear.
  • Two collinear vectors of the same magnitudes but opposite directions are said to be opposite vectors.
  • A vector that is opposite toVectors in Plane | Engineering Mathematics - Civil Engineering (CE)is denoted as Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)shows the above right figure.
  • Three or more vectors are said to be coplanar if they lie on the same plane. If two of three vectors are collinear then these vectors are coplanar.
  • To prove this statement, take vectors, Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)of whichVectors in Plane | Engineering Mathematics - Civil Engineering (CE)are collinear.
  • By using translation bring the tails of all three vectors at the same point. Then, the common line of vectors, Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)and the line in which lies the vector Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)determine the unique plane.
  • Therefore, if vectors are parallel to a given plane, then they are coplanar.

Addition of Vectors

  • The sum of vectors,Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)can be obtained graphically by placing the tail of Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)to the tip or head of Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) using translation. Then, draw an arrow from the initial point (tail) of Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) to the endpoint (tip) of Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)to obtain the result.

Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

  • The parallelogram in the above figure shows the Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) where, to the tip ofVectors in Plane | Engineering Mathematics - Civil Engineering (CE) by translation, placed addition is the tail ofVectors in Plane | Engineering Mathematics - Civil Engineering (CE) then, drawn is the resultant  Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) by joining the tail of Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) to the tip of Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)
    Note that the tips of the resultant and the second summand should coincide.

Thus, in the above figure shown is, the triangle rule (law) and the parallelogram rule for finding the resultant or the addition of the two given vectors. The result is the same vector, that is

Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)Therefore, vector addition is commutative.

Since vectors, Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) form a triangle, they lie on the same plane, meaning they are coplanar.
Addition of three vectors,Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) is defined as Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) and represented graphically
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)The above diagrams show that vector addition is associative, that is
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)The same way defined is the sum of four vectors.
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)If by adding vectors obtained is a closed polygon, then the sum is a null vector.
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

By adding a vector Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) to its opposite vector Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) , graphically it leads back to the initial point, therefore Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)  so, the result is the null vector.

Subtraction of Vectors

Subtraction of two vectors, Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) is defined as addition of vectors Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) that is, 

  • Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)As shows the right figure, subtraction of two vectors can be accomplished directly. 
  • By using translation place tails of both vectors at the same  point and connect their tips.
  • Note that the arrow (tip) of the difference coincides with the tip of the first vector (minuend).

Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

Scalar Multiplication or Multiplication of a Vector by Scalar


Scalar is a quantity which is fully expressed by its magnitude or size like length, time, mass, etc. as any real number. By multiplying a vector a by a real number λ obtained is the vector λ a  collinear to a but,   λ times longer than  a  if  | λ | > 1,   or    shorter than  a  if  | λ | < 1, and directed as  a  if  λ> 0,    or    opposite to  a  if  λ < 0, as is shown in the below figure.
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)Thus, the magnitude of the vector λ a equals to the product of the absolute value of the real number λ and the magnitude of the vector a, that is
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)Besides, for the multiplication of a vector by a real number following rules hold:

  1. λ· ( a + b ) = λab
  2.  ( λ + μ ) · a  = λ a + μ a,   λ, μ ∈ R
  3. λ ( μ a )  = μ( λ a ) = ( μλ ) a
  4. 1 · a  =  a,     -1 · a  = - a
  5. 0 · a  =  0,       μ · 0  = 0

Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)In the similar triangles ABC and ADE in the right
figure, Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)
therefore, AE = λ·  AC.
Since         AE = λ a + λ b    and    AC = a + b
then,          λ a + λ b = λ · ( a + b ).
Vectors in Plane | Engineering Mathematics - Civil Engineering (CE)

The document Vectors in Plane | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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