Revision: 11th Std >> Scalars and Vectors MAH-MHT CET (PCM/PCB) Scalars and Vectors

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 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.

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 product of the magnitudes of two vectors and the sine of the angle between them, giving a vector quantity perpendicular to the plane of both vectors, is called the vector or cross product.

The product of the magnitudes of two vectors and the cosine of the angle between them, giving a scalar quantity, is called the scalar or dot product.

“Calculus is the study of continuous (not discrete) changes in mathematical quantities.”

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

Scalar (Dot) Product:

• A ⋅ B = AB cos θ = A_x​B_x​ + A_y​B_y​ + A_z​B_z​
• Commutative:  A ⋅ B = B ⋅ A
• Distributive over addition: A ⋅ (B + C) = A ⋅ B + A ⋅ C
• Geometric interpretation: Product of the magnitude of one vector by the component of the other in the direction of the first
• A ⋅ A = A²
• If A ⊥ B, then A ⋅ B = 0

Vector (Cross) Product:

• A × B = (ABsinθ)\[\hat n\] = ​|\[\hat i\] \[\hat j\] \[\hat k\] A_x A_y A_z B_x B_y B_z|​​​
• Not commutative: A × B ≠ B × A
• Distributive over addition: A × (B + C) = A × B + A × C
• Geometric interpretation: Magnitude equals the area of the parallelogram whose adjacent sides are the two co-initial vectors
• A × A = 0
• If A ∥ B, then A × B = 0