Definitions [31]
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.
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 physical quantity that is described with magnitude alone is called a scalar.
When a vector \[\vec P\] is split into two mutually perpendicular parts along the horizontal and vertical axes, those parts are called rectangular components.
Vectors that act in the same plane are called coplanar vectors.
Vectors that are perpendicular to each other are called orthogonal vectors.
A vector is any quantity that needs both magnitude (size) and direction to be completely described.
OR
The physical quantities which have both magnitude and direction, obey the laws of vector addition, and are specified by a number with a unit and its direction (e.g., displacement, velocity, force, momentum) are called vector quantities or vectors.
A vector having the same magnitude as the original vector but having an opposite direction is called the negative of a vector.
The length or the magnitude of a vector is called the modulus of a vector.
A vector of unit magnitude drawn in the direction of a given vector is called a unit vector.
A vector that has zero magnitude and an arbitrary direction, represented by \[\vec 0\], is called a zero vector or null vector.
The vectors which act in the same plane are called co-planar vectors.
Vectors that are perpendicular to each other are called orthogonal vectors.
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.
A physical quantity that is described with magnitude alone is called a scalar.
When a vector \[\vec P\] is split into two mutually perpendicular parts along the horizontal and vertical axes, those parts are called rectangular components.
Vectors that act in the same plane are called coplanar vectors.
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.
A physical quantity that is described with both magnitude and direction is called a vector.
When a vector is resolved into components along mutually perpendicular directions (like x and y axes in 2D, or x, y, and z axes in 3D), these components are called rectangular or Cartesian components.
OR
When a vector is resolved along two mutually perpendicular directions, the components so obtained are called rectangular components of the given vector.
The three mutually perpendicular unit vectors \[\hat i\], \[\hat j\], \[\hat k\] used in three-dimensional space to describe the direction of any vector — where \[\hat i\] is along X-axis, \[\hat j\] along Y-axis, and \[\hat k\] along Z-axis — are called an orthogonal triad of base vectors.
A vector \[\vec V\] can be expressed as the sum of two or more vectors along fixed directions. This process is known as vector resolution.
OR
The process of splitting a single vector into two or more vectors in different directions which together produce same effect as produced by the single vector alone is called resolution of vector.
The splitting vectors obtained when a single vector is resolved into two or more vectors in different directions are called component vectors.
The values of cosα, cosβ, and cosγ which are the cosines of the angles subtended by the rectangular components with the given vector are called direction cosines of a vector.
Formulae [12]
If two vectors \[\vec P\] and \[\vec Q\] act at an angle θ, the magnitude of their resultant is:
| Condition | Angle | Resultant |
|---|---|---|
| Parallel vectors | 0° | R = P + Q |
| Perpendicular vectors | 90° | R = \[\sqrt{P^{2}+Q^{2}}\] |
| Anti-parallel vectors | 180° | R = P − Q |
If a vector \[\vec P\] is resolved into two rectangular components:
- Horizontal component: Px = P cos θ
- Vertical component: Py = P sin θ
\[\vec P\] ⋅ \[\vec Q\] = PQ cos θ
| θ | Dot Product |
|---|---|
| 0° | PQ |
| 90° | 0 |
| 180° | −PQ |
∣\[\vec P\] × \[\vec Q\]∣ = PQ sin θ
| θ | Cross Product |
|---|---|
| 0° | 0 |
| 90° | PQ |
| 180° | 0 |
∣\[\vec P\] × \[\vec Q\]∣ = PQ sin θ
| θ | Cross Product |
|---|---|
| 0° | 0 |
| 90° | PQ |
| 180° | 0 |
If two vectors \[\vec P\] and \[\vec Q\] act at an angle θ, the magnitude of their resultant is:
| Condition | Angle | Resultant |
|---|---|---|
| Parallel vectors | 0° | R = P + Q |
| Perpendicular vectors | 90° | R = \[\sqrt{P^{2}+Q^{2}}\] |
| Anti-parallel vectors | 180° | R = P − Q |
If a vector \[\vec P\] is resolved into two rectangular components:
- Horizontal component: Px = P cos θ
- Vertical component: Py = P sin θ
\[\vec P\] ⋅ \[\vec Q\] = PQ cos θ
| θ | Dot Product |
|---|---|
| 0° | PQ |
| 90° | 0 |
| 180° | −PQ |
When a vector \[\vec A\] is resolved into three-dimensional rectangular components, it is given by:
The magnitude of vector \[\vec A\] resolved into three-dimensional components is:
A = \[\sqrt{A_x^2+A_y^2+A_z^2}\]
If α, β, and γ are the angles subtended by the rectangular components with the given vector, then:
cos α = \[\frac {A_x}{A}\], cos β = \[\frac {A_y}{A}\], cos γ = \[\frac {A_z}{A}\]
The sum of squares of all direction cosines is always equal to 1:
cos2α + cos2β + cos2γ = 1
Theorems and Laws [12]
If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, then their sum is \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]
If two co-initial vectors are represented in magnitude and direction by adjacent sides of a parallelogram, then their sum is represented in magnitude and direction by the diagonal of the parallelogram passing through the common point.
∴ \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]
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).
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.
The resultant R of two vectors \[\vec P\] and \[\vec Q\] always lies between their difference and their sum:
If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, then their sum is \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]
If two co-initial vectors are represented in magnitude and direction by adjacent sides of a parallelogram, then their sum is represented in magnitude and direction by the diagonal of the parallelogram passing through the common point.
∴ \[\overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}}\]
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).
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.
The resultant R of two vectors \[\vec P\] and \[\vec Q\] always lies between their difference and their sum:
