Scalars and Vectors quantities

• A vector quantity is that which has both magnitude direction and satisfies laws of vector addition, a scalar quantity is that which has only magnitude.
• A unit vector is that which has 1 unit magnitude and a null vector is that which has O magnitude \hat{r} = \frac{\overline{r}}{|r|}
• A vector-quantity can be resolved in to any number of components when resolved in to two components Rx = R cos θ and Ry = R sin θ
• A vector resolved in to 3D space then the direction cosines as \cos \alpha = \frac{R_{x}}{|R|}, \cos \beta = \frac{R_{y}}{|R|}, \cos \gamma = \frac{R_{z}}{|R|} and cos2 α + cos2 β + cos2 γ = 1.
• Parallelogram law of vectors states that  if two vectors are represented both in magnitude and direction by the two sides of a parallelogram from the same point then their resultant is represented by the diagonal passing through the same point both in magnitude and direction.
R = \sqrt{P^{2} + Q^{2} + 2 PQ \cos \theta}
• Triangle law of vectors states that, if two vectors are represented in both magnitude and direction by the two sides of a triangle taken in order then their resultant is represented by the closing side of a triangle taken in the reverse order.
• The dot product of two vectors is the product of magnitudes of the vectors and cos of angle between them. \overline{a} \cdot \overline{b} = |a||b|\cos \theta Dot product is used to find the angle between the vectors.
• The cross product of two vectors is the product of magnitude of vectors and sin of angle between them. \overline{a} \times \overline{b} = |a||b|\sin \theta \cdot \hat{n}
• Cross product of two vectors is not commutative cross product obeys distributive \overline{A} \times \left(\overline{B} + \overline{C}\right) = \overline{A} \times \overline{B} + \overline{A} \times \overline{C} If vectors are parallel then cross product = 0
• Dot product of two vectors is Commutative Dot product obeys Destributive \overline{A} \cdot \left(\overline{B} + \overline{C}\right) = \overline{A} \cdot \overline{B} + \overline{A} \cdot \overline{C} If vectors are perpendicular the Dot product = 0

View the Topic in this video From 2:55 To 55:46

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1. A unit vector in the direction of vector A is given by
\hat{A} = \frac{A}{A}

2. The displacement vector for AB is Δr = r2r1

3. Triangle Law of Vectors Addition: If two vectors A and B acting at a point are inclined at an angle θ, then their resultant
R = \sqrt{A^{2} + B^{2} + 2AB \cos \theta}

4. If the resultant vector R subtends an angle β with vector A, then
\tan \beta = \frac{B \sin \theta}{A + B \cos \theta}

5. Parallelogram Law of Vectors Addition: Resultant of vectors A and B is given by
R = \sqrt{A^{2} + B^{2} + 2AB \ \cos \ \theta}

6. If the resultant vector R subtends an angle β with vector A, then
\tan \beta = \frac{B \sin \theta}{A + B \cos \theta}

7. Polygon Law of Vectors Addition: R = A + B + C + D + E

8. OE = OA + AB + BC + CD + DE

(i) Vector addition is commutative, i.e. A + B = B + A

(ii) Vector addition is associative, i.e., A + (B + C) = B + (C + A) = C + (A + B)

(iii) Vector addition is distributive, i.e. m (A + B) = mA + mB

(iv) A + 0 = A

10. Resolution of Vectors into two Components: Suppose OP = λA and PQ = μB, where λ and μ are two real numbers.

The resultant vector, R = λA + μB

11. Resolution of a Vector into Rectangular Components: Magnitude of vector A = \sqrt{A_{x}^{2} + A_{y}^{2}}

12. Direction Cosines of a Vector: cos2 α + cos2 β + cos2 γ = 1

13. Subtraction of Vectors: AB = A + (−B)

14. Scalar or Dot Product of Two Vectors: A · B = AB cos θ

15. Properties of Scalar Product:

(i) Scalar product is commutative, i.e., A · B = B · A

(ii) Scalar product is distributive, i.e., A · (B + C) = A · B + A · C

(iii) Scalar product of two perpendicular vectors is zero A · B = AB cos 90° = 0

(iv) Scalar product of two parallel vectors or anti-parallel vectors is equal to the product of their magnitudes, i.e.
A · B = AB cos 0° = AB (for parallel)
A · B = AB cos 180° = −AB (for anti-parallel)

(v) Scalar product of a vector with itself is equal to the square of its magnitude, i.e. A · A = AA cos 0° = A2

(vi) Scalar product of orthogonal unit vectors
\tt \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1
and            \tt \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0

(vii) Scalar product in cartesian coordinates
\tt A \cdot B = (A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}) \cdot (B_{x} \hat{i} + B_{y} \hat{j} + B_{z} \hat{k}) = A_{x}B_{x} + A_{y}B_{y} + A_{z}B_{z}

16. Vector or Cross Product of Two Vectors: \tt A \times B = AB \sin \theta \ \hat{n}

17. Properties of Vector Product:

(i) Vector product is not commutative, i.e. \tt A \times B \neq B \times A \ [\therefore (A \times B) = - (B \times A)]

(ii) Vector product is distributive, i.e. A × (B + C) = A × B + A × C

(iii) Vector product of two parallel vectors is zero, i.e., A × B = AB sin 0° = 0

(iv) Vector product of any vector with itself is zero. A × A = AA sin 0° = 0

(v) Vector product of orthogonal unit vectors

\tt \hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0
and \tt \hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}