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

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."

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

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

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

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

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

If two vectors can be represented both in magnitude and direction by the two sides of a triangle taken in the same order, then their resultant is represented both in magnitude and direction by the third side of the triangle taken in the opposite order — this is called the Triangle Law of Vector Addition.

If two vectors can be represented both in magnitude and direction by the two adjacent sides of a parallelogram drawn from a common point, then their resultant is completely represented, both in magnitude and direction, by the diagonal of the parallelogram passing through that point — this is called the Parallelogram Law of Vector Addition.

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.

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.

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