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.

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.

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.

The scalar product or dot product of two nonzero vectors \[\vec P\] and \[\vec Q\] is defined as the product of the magnitudes of the two vectors and the cosine of the angle θ between the two vectors.

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

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:

\[\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|\cos\theta=AB\cos\theta\]

• 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!