Home | About Mathguru | Advertisements | Teacher Zone | FAQs | Contact Us | Login

If you like what you see in Mathguru
Subscribe Today
For 12 Months
US Dollars 12 / Indian Rupees 600
Available in 20 more currencies if you pay with PayPal.
Buy Now
No questions asked full moneyback guarantee within 7 days of purchase, in case of Visa and Mastercard payment

Example:Solve to Find an Unknown Vector when Two Vectors are Equal.

Post to:

Bookmark and Share



Euclidean vector


Illustration of a vector


A vector going from A to B

In elementary mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or - as here - simply a vector) is a geometric object that has both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by 


A vector is a geometric entity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction. In rigorous mathematical treatments, a vector is defined as a directed line segment, or arrow, in a Euclidean space. When it becomes necessary to distinguish it from vectors as defined elsewhere, this is sometimes referred to as a geometric, spatial, or Euclidean vector.

As an arrow in Euclidean space, a vector possesses a definite initial point and terminal point. Such a vector is called a bound vector. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector. Thus two arrows  and  in space represent the same free vector if they have the same magnitude and direction: that is, they are equivalent if the quadrilateral ABB′A′ is a parallelogram. If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.

The term vector also has generalizations to higher dimensions and to more formal approaches with much wider applications.



Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, as a. (Uppercase letters are typically used to represent matrices.) Other conventions include  or a, especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as  or AB.

Vectors are usually shown in graphs or other diagrams as arrows (directed line segments), as illustrated in the figure. Here the point A is called the origin, tail, base, or initial point; point B is called the head, tip, endpoint, terminal point or final point. The length of the arrow is proportional to the vector's magnitude, while the direction in which the arrow points indicate the vector's direction.

As an example in two dimensions (see figure), the vector from the origin O = (0,0) to the point A = (2,3) is simply written as


A vector in the Cartesian plane, showing the position of a point A with coordinates (2, 3).

The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation  is usually not deemed necessary and very rarely used.

In three dimensional Euclidean space (or), vectors are identified with triples of scalar components:

also written

These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows:

In introductory physics textbooks, the standard basis vectors are often instead denoted  (or, in which the hat symbol ^ typically denotes unit vectors). In this case, the scalar and vector components are denoted ax, ay, az, and axayaz. Thus,



Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors


are equal if

(Our solved example in mathguru.com uses this concept)




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.