Revision: Vector Algebra Mathematics HSC Science Class 11 Tamil Nadu Board of Secondary Education

The magnitude of vector \[\vec{AB}\] is the length of the directed line segment AB. It is written as \[|\vec{AB}|\], \[|\vec{a}|\], or simply a. The magnitude of a vector is never negative because it represents length.

In three-dimensional geometry, the vector drawn from the origin O(0, 0, 0) to a point P(x, y, z) is called the position vector of the point P. It is written as \[\vec{OP}\]. If point P(x, y, z) is given, then the magnitude of its position vector is:

A scalar quantity is a physical quantity that has magnitude only.

A vector quantity is a physical quantity that has magnitude as well as direction.

A vector is a quantity that has magnitude as well as direction. Geometrically, a vector is represented by a directed line segment such as  \[\vec{AB}\], where A is the initial point and B is the terminal point.

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.

A vector having the same magnitude as the original vector but having an opposite direction is called the negative of a vector.

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 of unit magnitude drawn in the direction of a given vector is called a unit vector.

The length or the magnitude of a vector is called the modulus of a vector.

In three-dimensional geometry, the vector drawn from the origin O(0, 0, 0) to a point P(x, y, z) is called the position vector of the point P. It is written as \[\vec{OP}\]. If point P(x, y, z) is given, then the magnitude of its position vector is:

A scalar quantity is a physical quantity that has magnitude only.

A vector quantity is a physical quantity that has magnitude as well as direction.

A vector is a quantity that has magnitude as well as direction. Geometrically, a vector is represented by a directed line segment such as  \[\vec{AB}\], where A is the initial point and B is the terminal point.

The magnitude of vector \[\vec{AB}\] is the length of the directed line segment AB. It is written as \[|\vec{AB}|\], \[|\vec{a}|\], or simply a. The magnitude of a vector is never negative because it represents length.

The splitting vectors obtained when a single vector is resolved into two or more vectors in different directions are called component 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 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.

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 values of cos⁡α, cos⁡β, 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.

The cosines of these angles are called the direction cosines of the line.

So, the direction cosines are written as (l, m, n).

If a directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively, then these are called the direction angles of the line.

Any three numbers proportional to the direction cosines of a line are called the direction ratios of the line.

If (a, b, c) are direction ratios, then:

The sum of squares of all direction cosines is always equal to 1:

cos²α + cos²β + cos²γ = 1

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 magnitude of vector \[\vec A\] resolved into three-dimensional components is:

A = \[\sqrt{A_x^2+A_y^2+A_z^2}\]​​

When a vector \[\vec A\] is resolved into three-dimensional rectangular components, it is given by:

• Scalars have only magnitude.

• Vectors have magnitude and direction.

• Vectors are represented by directed line segments.

• \[\vec{AB}\] represents a vector from A to B.

• Magnitude of a vector is its length and is always non-negative.

• \[\vec{OP}\] is the position vector of point \[P(x, y, z)\].

• \[|\vec{OP}| = \sqrt{x^2 + y^2 + z^2}\].

• Distance vs Displacement: Distance (5 km) is scalar; displacement (5 km north) is vector.
• Speed vs Velocity: Speed (60 km/h) is scalar; velocity (60 km/h north) is vector.
• Vectors add differently: You cannot simply add vectors like scalars. A 5 N force east + 5 N force north ≠ 10 N!

• Scalars have only magnitude.

• Vectors have magnitude and direction.

• Vectors are represented by directed line segments.

• \[\vec{AB}\] represents a vector from A to B.

• Magnitude of a vector is its length and is always non-negative.

• \[\vec{OP}\] is the position vector of point \[P(x, y, z)\].

• \[|\vec{OP}| = \sqrt{x^2 + y^2 + z^2}\].

• Direction Cosines (DCs) of a line: \[(l, m, n) = (\cos \alpha, \cos \beta, \cos \gamma)\].

• Main identity: \[l^2 + m^2 + n^2 = 1\].

• Direction Ratios (DRs): Are proportional to DCs.

• Relation between DRs and DCs: If DRs are (a, b, c), then DCs are proportional to (a, b, c) divided by \[\sqrt{a^2 + b^2 + c^2}\].

• DRs for two points: For points \[P(x_1, y_1, z_1)\] and \[Q(x_2, y_2, z_2)\], DRs are \[(x_2 - x_1, y_2 - y_1, z_2 - z_1)\].