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VECTORS

What is a vector?

A vector is a quantity that has both direction and magnitude. A vector is denoted by a directed line segment. The magnitude of a vector is the length of the segment and the direction of a vector is where the segment is pointing. Vectors are named like segments or by small letters.

A vector from a point A to a point B is denoted , and a vector v may be denoted , or more commonly, v. The point A is often called the “tail” of the vector, and B is called the vector’s “head.” A vector with unit length is called a unit vector and is denoted using a hat, . An arbitrary vector may be converted to a unit vector by dividing by its norm (i.e., length),

and

Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object; such diagrams are known as free-body diagrams. An example of a scaled vector diagram is shown in the diagram at the right to depict a displacement vector. Observe that there are several characteristics of this diagram which make it an appropriately drawn vector diagram.

• a scale is clearly listed

• an arrow (with arrowhead) is drawn in a specified direction; thus, the vector has a head and a tail.

• the magnitude and direction of the vector is clearly labeled; in this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).

Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector is often called a two-vector,an n-dimensional vector is often called an n-vector, and so on.

A vector that has no magnitude but has a direction is called a zero vector.

Vectors are equal when their magnitudes and direction are equal.

ADDITION OF VECTORS

The Parallelogram Rule

Given two vectors, c and d, not equal to each other:

place the initial points of both vectors in the same point and call it point O. Draw a line parallel to vector c at the endpoint of vector d and also draw a line parallel to vector d at the endpoint of vector c, name the point of intersection of the parallel lines, call it point B.

Head to Tail Rule:

Given two unequal vectors

Place the initial point of one vector at endpoint of the other vector. place a triangle from the joined vectors and the third vector is the vector sum of the two vectors.

If two vectors are having the same direction but different magnitude, their vector sum is the sum of the magnitudes in the same direction.

If two vectors are having the same magnitude and are in opposite directions their vector sum is a zero vector.

VECTOR SUBTRACTION

If two vectors are in opposite direction and have different magnitudes, then their vector sum is the difference in magnitudes of the vectors in the direction of the vector with the larger magnitude.