Introduction to Vectors

A physical quantity that is described with magnitude alone is called a scalar.

A physical quantity that is described with both magnitude and direction is called a vector.

A vector whose magnitude is zero is called a zero vector.

A vector that has the same magnitude as a given vector but acts in the opposite direction is called a negative vector.

Two vectors having the same magnitude and the same direction are called equal vectors.

A vector that describes the position of a point with respect to the origin is called a position vector.

A vector that has a magnitude of one unit and is used to indicate direction is called a unit vector.

Vectors that act in the same plane are called coplanar vectors.

Vectors that are perpendicular to each other are called orthogonal vectors.

When a vector \[\vec P\] is split into two mutually perpendicular parts along the horizontal and vertical axes, those parts are called rectangular components.

For any two vectors \[\vec P\] and \[\vec Q\]​:

The commutative law holds true for addition of vectors but not for subtraction.

For three vectors \[\vec P\], \[\vec Q\]​, and \[\vec R\]:

The associative law holds true for addition of vectors but not for subtraction.

Two vectors can be added using either the Triangle Law or the Parallelogram Law. When two vectors \[\vec P\] and \[\vec Q\]​ are represented as two sides of a triangle (or two adjacent sides of a parallelogram), their resultant is represented by the third side (or the diagonal).

The resultant R of two vectors \[\vec P\] and \[\vec Q\]​ always lies between their difference and their sum:

If two vectors \[\vec P\] and \[\vec Q\]​ act at an angle θ, the magnitude of their resultant is:

If a vector \[\vec P\] is resolved into two rectangular components:

• Horizontal component: P_x = P cos⁡ θ
• Vertical component: P_y = P sin ⁡θ

\[\vec P\] ⋅ \[\vec Q\] ​= PQ cos θ

∣\[\vec P\] × \[\vec Q\]​∣ = PQ sin θ