Science Library - free educational site

Vector basics

Vectors are lines with direction and magnitude from a point. This point could be the origin (0, 0), or the end of another vector.

Vector question

Column Vector Form

In the diagram, an object moves from the origin (0, 0) to point B. It does not follow a straight line, but we are only interested in its displacement from the origin. So, we can say it arrives at B by moving 5 squares along the x-axis, and 5 squares up the y-axis. We can write this as a column vector:

${AB}↖{→} = (\table (x_f - x_i);(y_f - y_i)) $

$= (\table 5-0;5-0) = (\table 5;5)$

where $x_f$ and $y_f$ are the final x and y positions, and $x_i$ and $y_i$ are the intial x and y positions.

Parametric Equation Form

If the length of a side of one square is a 'unit', then we can say that the object has moved 5 units along the x-axis, and 5 units up the y-axis. One unit in the x-axis we call 'i', and one unit in the y-axis we call 'j'. This allows us to write the same vector with the parameters i and j:

${AB}↖{→} = 5i + 5j$

Vector question


Write the path of the object in the diagram in column vector and parametric equation forms.

Column vector: ${AB}↖{→} = (\table (x_f - x_i);(y_f - y_i)) $

$= (\table 4-1.5;1-5) = (\table 2.5;-4)$

Parametric equation: ${AB}↖{→} = 2.5i - 4j$

Unit Vector

A unit vector is a vector that is one unit long. A unit vector in the x-direction is 1i, and a unit vector in the y-direction is 1j. Therefore, the vector a =[a1, a2] may also be written in the form: a1i + a2j.

The angle between two vectors is:

$$cosθ = {a1.b1 + a2.b2}/{(|u|⋅|v|)} $$

The dot product of two vectors is defined as:

$$ u.v = |u|⋅|v|⋅cosθ$$

The magnitude of a vector can be calculated using the fact that a vector can be broken down to two components, in the x and y direction.

Vector magnitude and angle
Vector magnitude and angle

The length of a vector is determined using the Pythagorean Theorem:

let a be a vector a =[a1, a1], then the length is |a| = √a12 + a22.

Vectors in Real Life

Vectors describe quantities which can have magnitude and direction. These may be:

A hanging object has all forces in equilibrium
Vectors indicate the magnitude and direction of quantities, such as the forces of weight and tension in rope. In equilibrium, the sum of these force vectors equals zero.
  • force
  • displacement
  • velocity
  • acceleration
  • momentum

For example, a weight suspended by two ropes has forces in equilibrium. these means there is no net force acting, so the mass does not accelerate in any direction. The forces may be drawn in a free-body diagram, where the relative strengths of the forces are indicated by the lengths of the vector, as well as their direction. if the force vectors are added together, they will result in a zero vector.

Content © Andrew Bone. All rights reserved. Created : February 12, 2014

Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:


and many more articles in the subject:

Subject of the Week


Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on

Gravity lens

Great Scientists

Tycho Brahe

1546 - 1601

Tycho Brahe was a Danish astronomer, whose meticulous observations and ideas about the nature of planets, stars and comets, although erroneously accommodating rather than directly challenging the long-standing Aristotelian and Ptolemaic versions of the heavens, they were nevertheless an important contribution to the coming scientific revolution, and specifically enabled his assistant, Johannes Keppler, to derive his breakthrough laws of planetary motion.

Tycho Brahe
IT Forum by Sean Bone

Quote of the day...

The great thing about science is that it is true whether you believe in it or not.

ZumGuy Internet Promotions

English language school