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

The Vector Product (or Cross Product) is a method of multiplying two vectors (\[\vec P\] and \[\vec Q\]) that results in a new vector (\[\vec R\]). This new vector is fundamentally related to the rotation or perpendicular effects created by the two original vectors.

The magnitude of the resulting vector R is defined by the product of the magnitudes of the two vectors and the sine of the smaller angle (θ) between them.
Magnitude: ∣R∣ = ∣ P × Q ∣ = PQ sin θ

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

"dy/dx is called the derivative of y with respect to x (which is the rate of change of y with respect to change in x) and the process of finding the derivative is called differentiation."

F(x) = ∫f(x) dx is called the indefinite (without any limits on x) integral of f(x).

The representation \[\int_{x=a}^{x=b}\] f(x)dx is called the definite integral of f(x) from x = a to x = b.

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:

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

\[\vec{A}\times\vec{B}=AB\sin\theta\hat{n}\]

If a number of vectors are represented both in magnitude and direction by the sides of an open polygon taken in the same order, then their resultant is represented both in magnitude and direction by the closing side of the polygon taken in opposite order — this is called the Polygon Law of Vector Addition.

• 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!

1. Component Method: Resultant R = A + B is found as R_x = A_x + B_x​, R_y = A_y + B_y, R_z = A_z + B_z​, giving R = R_x\[\hat i\] + R_y\[\hat j\] + R_z\[\hat k\].

2. Laws of Addition: Triangle law (head-to-tail), Parallelogram law (tail-to-tail, diagonal = resultant), and Polygon law (for multiple vectors, closing side = resultant).

3. Magnitude (Addition): When A and B are at angle θ, R = \[\sqrt{A^2+B^2+2AB\cos\theta}\].

4. Magnitude (Subtraction): Change the sign to minus — ∣R∣ = ​\[\sqrt{A^2+B^2-2AB\cos\theta}\].

5. Direction of Resultant: tan⁡α = \[\frac{B\sin\theta}{A+B\cos\theta}\] for addition; tan⁡β = \[\frac{B\sin\theta}{A-B\cos\theta}\] for subtraction.