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 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  is 

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

A vector with no magnitude, i.e., if the tail and the head coincide, is called the zero or null vector denoted

Collinear, Opposite and Coplanar Vectors

Vectors are said to be collinear if they lie on the same line or on parallel lines.

• Vectors,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 tois denoted as 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, of whichare collinear.
• By using translation bring the tails of all three vectors at the same point. Then, the common line of vectors, and the line in which lies the vector determine the unique plane.
• Therefore, if vectors are parallel to a given plane, then they are coplanar.

Addition of Vectors

• The sum of vectors,can be obtained graphically by placing the tail of to the tip or head of  using translation. Then, draw an arrow from the initial point (tail) of  to the endpoint (tip) of to obtain the result.

• The parallelogram in the above figure shows the  where, to the tip of by translation, placed addition is the tail of then, drawn is the resultant   by joining the tail of  to the tip of 
  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

Therefore, vector addition is commutative.

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

By adding a vector  to its opposite vector  , graphically it leads back to the initial point, therefore   so, the result is the null vector.

Subtraction of Vectors

• 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).

Scalar Multiplication or Multiplication of a Vector by Scalar

1. λ· ( a + b ) = λa +λ b
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

In the similar triangles ABC and ADE in the right
figure,
therefore, AE = λ·  AC.
Since         AE = λ a + λ b    and    AC = a + b
then,          λ a + λ b = λ · ( a + b ).