Scalars and Vectors PDF

# Vectors and Scalars All the physical quantities are classified into two groups on the basis of magnitude and direction. * **Scalars**: Those physical quantities having only magnitude are called scalars. - *Examples*: mass, time, distance, speed, area, work, energy, etc. * **Vectors**: Those p...

# Vectors and Scalars All the physical quantities are classified into two groups on the basis of magnitude and direction. * **Scalars**: Those physical quantities having only magnitude are called scalars. - *Examples*: mass, time, distance, speed, area, work, energy, etc. * **Vectors**: Those physical quantities having magnitude as well as direction are called vectors. - *Examples*: Displacement, Velocity, acceleration, force, Momentum etc. ## Notation of a Vector A vector is denoted by an alphabet giving an arrow (→) over its head. - *Velocity is denoted as V*, *force is denoted as F*, *momentum is denoted as P*. ## Representation of Vectors A vector is represented by a directed line segment i.e a straight line drawing an arrow at one end. - **The length of the line indicates magnitude.** - **The arrow sign indicates direction.** ## Modulus of a Vector Modulus of a vector gives only magnitude. Modulus of a vector is denoted by enclosing it between two vertical parallel lines. - *Modulus of a = |a| = √a² = a*. ## Classification of Vectors * **Unit Vector**: A vector whose modulus is equal to 1 is called a unit vector. - A unit vector is denoted by a small letter and the arrow (→) is replaced by a caret or hat. - *|â| = 1* * **Null Vector**: A vector whose modulus is zero is called a null vector. - *Example*: |a|=0; a= null vector - *Velocity of a body projected vertically upward at the maximum height = 0; |v|=0* - *Resultant force of two equal and opposite forces is a null vector* * **Proper Vector**: A vector whose modulus is not equal to zero. - *|a|≠0* * **Like Vectors**: Two or more vectors are called like vectors if they have the same direction and different magnitudes. * **Unlike Vectors**: Two or more vectors are called unlike vectors if they have opposite direction and different magnitudes. * **Equal Vectors**: Two or more vectors are called equal vectors if they have the same magnitude and the same direction. * **(-ve) Vectors**: Two vectors are called (-ve) vectors of each other if they have the same magnitude, but opposite directions * **Co-Linear Vectors**: Two or more vectors are called co-linear vectors if they lie in the same line. * **Co-Planner Vectors**: Two or more vectors are called co-planner vectors if they lie in the same plane. * **Co-Initial Vectors**: Two or more vectors are called co-initial vectors if they have the same origin. * **Orthogonal Vectors**: Two or more vectors are called orthogonal vectors if they are perpendicular to each other. ## Addition of Two Vectors The head-and-tail rule: 1. **Join the head of a with the tail of b**. 2. **The vector joining the tail of a to the head of b gives the resultant vector: a+b = OA + AB = OB = c** ### General Addition of Vectors: - **AB + BC = AC** - **PB + BR = PR** - **AB + CB =** (This is not added). ### Addition of Several Vectors: - **Let OA = a, AB = b, BC = c, CD = d** - **a+b = OA + AB = OB** - **(a+b) + c = OB + BC = OC** - **(a+b+c) + d = OC + CD = OD** **Rules of addition of several vectors:** 1. **The resultant of several vectors is obtained by adding tail of the first vector to the head of the last vector** 2. **The resultant of several vectors in a closed figure is zero.** ## Commutative Law **Proof**: Let us consider a parallelogram ABCD. - **AD = a, AB = b, AC = c, BD = d** - **Since ABCD is a parallelogram, a + b = c, b + d = a** ## Associative Law - **(a+b)+c = a+(b+c)** ## Subtraction of Two Vectors - **a-b = a+(-b)** **To subtract b in a means to add -b in a** **Steps to subtract b in a:** 1. Draw -b by reversing the direction of b, keeping the same magnitude. 2. Join the head of a with the tail of -b. 3. The vector joining the tail of a to the head of -b gives the resultant vector: a+(-b) = a-b. ## Multiplication of Vectors ### Multiplication of a vector with a scalar When a vector is multiplied with a scalar, then the product is a vector quantity whose direction remains the same, but the magnitude becomes scalar times of the previous vector. - *2a = scalar* - *a = vector* - *2a = 2|a|a = vector* *Example*: - *m = mass of a body (scalar)* - *a = acceleration of the body (vector)* - *m x a = F = vector = force* ### Multiplication of two vectors If two vectors are multiplied then the product is either a scalar or a vector. Therefore, there are two types of products of two vectors. 1. **Scalar product or dot product**: a.b = |a| |b| cos θ = abcos θ 2. **Vector product or cross product**: a x b = |a| |b| sin θ n = ab sin θ n #### Properties of Scalar Product (Dot Product) 1. The dot product of two vectors is maximum if the angle between them is 0. i.e. two vectors are in the same direction. - *If θ=0, then cosθ=1 (maximum)* - *a.b = abcosθ = ab = |a| |b| maximum* 2. If two vectors are perpendicular to each other, then their dot product is 0. - *If two vectors are perpendicular to each other, then θ = 90°, so cosθ = 0* - *a.b = ab cos θ = ab cos 90 = 0* 3. Dot product of a vector with itself is the square of its modulus. - *a.a =|a|² = a²* 4. Dot product of two vectors is commutative. - *a . b = bacos θ = b.a* 5. Dot Product of two unit vectors: - *î.î = |î||î|cosθ = 1 x 1 x 1 = 1* - *Similarly, j.j = k.k = 1* - *î.j = |î||j|cos90 = 1 x 1 x 0 = 0; => î.j = 0* - *Similarly, j.k = k.i = 0* 6. Dot product between two vectors - *If a = a₁î + a₂j + a₃k* - *b = b₁î + b₂j + b₃k* - *Then, a.b = a₁b₁î.î + a₁b₂î.j + a₁b₃î.k + a₂b₁j.î + a₂b₂j.j + a₂b₃j.k + a₃b₁k.î + a₃b₂k.j + a₃b₃k.k* - *a.b = a₁b₁ + a₂b₂ + a₃b₃* *Example:* - *a = 3î + 2j - 2k* - *b = 2î + j – k* - *a.b = 6 + 2 + 2 = 10* *To find the angle between two vectors* - *a = 3î + 2j - 2k* - *b = 2î + j – k* - *a.b = |a| |b| cos θ* - *cos θ = (a.b)/(|a| |b|) = (6+2+2)/ (√(9+4+4)√(4+1+1))* - *cos θ = (10)/(√17√6) => θ ≈ 90°* #### Properties of Vector Product (Cross-Product) 1. *a x b = ab sin θ n* 2. *|a x b| = ab sin θ* 3. Cross product of two vectors is maximum if the angle between them is 90°. If two vectors are perpendicular to each other. - *If θ = 90°, then sin θ = 1* - *|a x b| = ab sin θ = ab* - *And the maximum is when a and b are perpendicular to each other* 4. Cross product of two vectors is 0 if the angle between them is 0°. i.e. two vectors are in the same direction - *If θ = 0°, then sin θ = 0 * - *|a x b| = ab sin θ = 0* 5. The cross product of a vector with itself is 0. - *a x a = aa sin θ n = 0* 6. The cross product of two vectors is anti-commutative. - *(a x b) = -(b x a)* 7. Cross product of two unit vectors. - *î x î = |î| |î| sin θ n = 1 x 1 x 0 n = 0* - *Similarly, j x j = k x k = 0* - *î x j = |î| |j| sin 90 n = 1 x 1 x 1 n = n* - *Similarly, j x k = n, k x i = n* - *î x i = -n* - *Similarly, j x i = -k, k x j = -i* 8. Cross product of two vectors. Let: - *a = a₁î + a₂j + a₃k* - *b = b₁î + b₂j + b₃k * Method 1: *a x b = a₁b₁îxî + a₁b₂îxj + a₁b₃îxk + a₂b₁jxî+ a₂b₂jxj + a₂b₃jxk + a₃b₁kxî + a₃b₂kxj + a₃b₃kxk * - *(0)î + (a₁b₃-a₃b₁)j + (a₂b₁-a₁b₂)k + (a₃b₂-a₂b₃)î + (0)j + (a₁b₂-a₂b₁)k* - *(a₃b₂-a₂b₃)î + (a₁b₃-a₃b₁)j + (a₂b₁-a₁b₂)k* Method 2: [1] [2] [3] *a x b = î (a₂b₃-a₃b₂) + j (a₃b₁-a₁b₃) + k (a₁b₂-a₂b₁)* *Example:* - *If a = 3î + 2j – 3k* - *b = - 2î + 2j + 5k* [1] [2] [3] *a x b = î (10 + 6) + j (-15 + 6) + k (6 + 4) = 16î - 9j + 10k* ## Triangle Law of Vector Addition If two vectors are represented by two sides of a triangle in magnitude and direction then their resultant is represented by the third side in magnitude, but direction is taken in reverse order. **Let us consider a ∆OAB** - **Let OA = a, AB = b and OB = c** - **a + b = OA + AB = OB = c** **So, c = √(a² + b² + 2ab cos θ) (magnitude)** **Direction of c:** - **Let OM represent a, MB represent b, and OB represents c** - **tan α = BM/OM = b sin θ/a + b cos θ** **α = tan^-1 (b sin θ/a + b cos θ) (Direction)** **Different Cases:** ### **Case 1: θ = 0** Two vectors are in the same direction. - *cosθ = 1 (maximum)* - *sinθ = 0* - *c = √(a² + b² + 2ab cos θ)* - *c = √(a² + b² + 2ab) = √(a + b)² = a + b* - *|c| = |a| + |b| = max'm* - *tan α = b sin θ/a + b cos θ = 0* - *α = 0* ### **Case 2: θ = 180°** Two vectors are in the opposite direction. - *cos θ = -1 (minimum)* - *sin θ = 0* - *c = √(a² + b² + 2ab cos θ) = √(a² + b² - 2ab) = √(a - b)² = a - b* - *|c| = |a| - |b| = minimum* - *If a = b then |c| = 0* - *tan α = b sin θ/a + b cos θ = 0* - *α = 0* ### **Case 3: θ = 90°** - *You would calculate the resultant vector* ## Resolving a vector into its two perpendicular components Let P be a point on the x-y plane. - **P = (x, y)** - **OP = r** - The position vector of P, r, is **r = xi + yj**. - The angle between r and the x-axis is θ. - **x = r cos θ** - **y = r sin θ** - **So, r = r cos θ i + r sin θ j** Let's apply these concepts to some examples: **Example 1: When two vectors are in the same direction** Say you have a car travelling 200m East and then 300m East. Both vectors are in the same direction. Therefore, the net displacement will be 200m + 300m = 500m East. **Example 2: When two vectors are in opposite directions** Suppose you have a car travelling 200m East and then 300m West. Now you will subtract the two displacements, and the net displacement will be 300m - 200m = 100m West. We make the displacement West because the 300m displacement vector is larger than the 200m vector. **Example 3: When two vectors are perpendicular to each other** Imagine an object moving 10m north and then 10m east. Here, we have two perpendicular vectors. To get the net displacement, we can use the Pythagorean theorem: Net displacement = √(10m² + 10m²) = √200m² = 14.1m **Example 4: When two vectors are at an angle to each other** Imagine two vectors of 10m each, with an angle of 60° between them. We can use the parallelogram law to find the resultant. This means: - The magnitude of the resultant is 10√3m - The direction of the resultant is 30° Keep in mind that the concepts of vectors and scalars are crucial in physics and other fields. Understanding them is crucial for analyzing and solving a wide range of problems that involve quantities that have both magnitude and direction.