[size=85][size=100][size=150]You must have heard that vectors are quantities which have magnitude and direction. Right? [br][br]But how to bring them into geometry? [br]Suppose we have a force of 4 N applied to a mass in the horizontal direction . How do we represent that in Geometry?[br]Let us consider a line segment of length 4 units. Will it suffice? No. We have to show the direction as well. How do we do that? [br]Suppose our line segment is AB and is starting at the point A and ending in B. How to show the direction now?Direction can be associated with some kind of movement. Right? [br]Notice that when we start to draw the line segment we start at A and move in a certain way. Why not move in the direction of the applied force, that is in the horizontal way to the right and end up in B after moving a distance of 4 units. [br]To show the motion in the manner as is indicated let us put an arrowhead at the end point B. [br][/size][/size][/size][size=150]This will certainly make sense.[/size]

[size=100][size=150]Two vectors will be equal if and only if their magnitude and directions match. Right ?[br]Suppose we have a vector [math]\vec{AB}[/math] and another vector [math]\vec{CD}[/math]. How can these two be equal?[br][/size][/size]

[size=150]A vector [math]\vec{AB}[/math] whose magnitude denoted by [math]\left|\vec{AB}\right|[/math] is equal to 1 is defined as a unit vector. Note that a unit vector may have any direction.[br]There are two special unit vectors denoted as [math]\vec{i}[/math] and [math]\vec{j}[/math] . The first one has a direction along positive x-axis and the second one along positive y-axis.[/size]

S[size=150]uppose we have a vector [math]\vec{AB}[/math] . Let [math]\alpha[/math] be any scalar then their product is denoted as [math]\alpha\vec{AB}[/math] which has the magnitude [math]\left|\alpha\right|\left|\vec{AB}\right|[/math] and has the same direction as [math]\vec{AB}[/math] if [math]\alpha\ge0[/math] and has opposite direction if [math]\alpha<0[/math].[/size]

[size=100][size=150]Observe how the value of a becomes the scalar multiple.[/size][/size]

[size=150]Any vector in a plane can be expressed as [math]\alpha\vec{i}+\beta\vec{j}[/math] where [math]\alpha,\beta[/math] are two scalars and [math]\vec{i}[/math] and [math]\text{\vec{j}}[/math] are the two unit vectors defined above.[/size]