## Motion in a Plane

# 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 R
_{x}= R cos θ and R_{y}= 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 cos
^{2}α + cos^{2}β + cos^{2}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 Δ**r** = **r**_{2} − **r**_{1}

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**

9. Properties of Vectors Addition:

(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**) = *m***A** + *m***B**

(iv) **A + 0 = A**

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

The resultant vector, **R** = λ**A** + μ**B**

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: cos^{2} α + cos^{2} β + cos^{2} γ = 1

13. Subtraction of Vectors: **A** − **B** = **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° = **A**^{2}

(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}