title: “Vectors - Recapitulation of lec1”

• Vectors are mathematical entities used to represent both magnitude and direction.

• They are denoted by a boldface letter or an arrow above a letter.

• Examples: a, b, c.

• Vectors can be represented in component form, i.e., along the x and y axes.

• Example: a = (a_x, a_y).

• Vector Addition: To add vectors, we add the corresponding components.

• Example: a + b = (a_x + b_x, a_y + b_y).

• The resultant vector is the vector sum of the given vectors.

• Example: r = a + b.

• Scalar Multiplication: A scalar can be multiplied with a vector.

• Example: ca = (c * a_x, c * a_y), where c is a scalar.

• The direction of the vector remains unchanged, but the magnitude is multiplied by the scalar.

• Vector Subtraction: To subtract vectors, we subtract the corresponding components.

• Example: a - b = (a_x - b_x, a_y - b_y).

• The resultant vector is the vector difference of the given vectors.

• Example: r = a - b.

• Magnitude of a Vector: The magnitude of a vector a is denoted by |a|.

• It is calculated using the Pythagorean theorem.

• Example: |a| = √(a_x² + a_y²).

• It represents the length or size of the vector.

• Unit Vector: A unit vector has a magnitude of 1 and is used to represent direction.

• It is denoted by a hat (^) above the vector.

• Example: â = (a_x/|a|, a_y/|a|).

• The unit vector points in the same direction as the original vector.

• Dot Product: The dot product of two vectors a and b is denoted by a · b.

• It is calculated by multiplying the corresponding components and summing the results.

• Example: a · b = a_x * b_x + a_y * b_y.

• The dot product gives a scalar value.

• Angle between Vectors: The angle θ between two vectors a and b is calculated using the dot product.

• Example: cos(θ) = (a · b) / (|a| * |b|).

• The angle is measured in radians or degrees, depending on the unit used.

• Cross Product: The cross product of two vectors a and b is denoted by a × b.

• It is calculated using the determinants of the vectors’ components.

• Example: a × b = (a_x * b_y - a_y * b_x) i.

• The cross product gives a vector perpendicular to the plane containing the given vectors.

• Applications of Vectors: Vectors have various applications in physics, engineering, and computer science.

• They are used to represent forces, velocities, displacements, electric fields, and more.

• Vector analysis is essential for understanding and solving real-life problems.

• Mastering vectors is crucial for success in many scientific and technical fields.

• Vector Projection: The projection of vector a onto vector b is denoted as proj_b a.
• It represents the component of a in the direction of b.
• The formula for projection is proj_b a = a · (b/|b|).
• Example: proj_b a = (a · b) / |b|.

• Vector Components: We can express a vector a as the sum of its components along the x and y axes.
• Example: a = a_x i + a_y j.
• The components represent the magnitude and direction of the vector along each axis.
• To find the components, we use trigonometry based on the angle θ.

• Vector Resolutions: Vector resolutions involve breaking down a vector into its component vectors.
• The horizontal component is given by a_x = |a| * cos(θ), where θ is the angle with the x-axis.
• The vertical component is given by a_y = |a| * sin(θ), where θ is the angle with the x-axis.
• The resolution helps analyze the vector’s effects in different directions.

• Vector Magnitude and Direction: The magnitude and direction of a vector can be determined from its components.
• The magnitude is given by |a| = √(a_x² + a_y²), using the Pythagorean theorem.
• The direction is determined by calculating the angle θ = tan^-1(a_y/a_x).
• Example: If a_x = 3 and a_y = 4, then |a| = 5 and θ ≈ 53.13°.

• Vector Equations: Vectors can be represented using equations to solve various problems.
• Example: If a + 2b = c, and a = 2i - j, b = i + 3j, find c.
• By substituting the given values, we can determine the expression for c.

• Angle between Vectors: The dot product can be used to find the angle between two vectors.
• By applying the cosine formula, we get cos(θ) = (a · b) / (|a| * |b|).
• Example: If a = 3i + 4j and b = 5i - j, then cos(θ) = (3 * 5 + 4 * -1) / (5 * √(9 + 16)).

• Scalar Projection: The scalar projection of a onto b is denoted as scalar_b a.
• It represents the magnitude of the projection of a onto b.
• The formula for scalar projection is scalar_b a = |a| * cos(θ).
• Example: scalar_b a = |a| * cos(θ) = |a| * (|a| * |b|) / (a · b).

• Collinear Vectors: Two vectors are said to be collinear if they lie on the same line.
• Collinear vectors have the property that one vector can be obtained by scalar multiplication of the other.
• Example: If a = kb, then a and b are collinear.
• Collinear vectors have the same or opposite directions.

• Coplanar Vectors: Three or more vectors are said to be coplanar if they lie in the same plane.
• Coplanar vectors can be represented by a triangle with vectors as sides.
• If the vectors are coplanar, then the determinant of the vectors’ components is zero.
• Example: If a, b, and c are coplanar, then |a x b · c| = 0.

• Vector Equality: Two vectors are considered equal if their corresponding components are equal.
• Example: If a = 3i + 4j and b = 3i + 4j, then a = b.
• Vector equality follows the principles of equality in algebra.

Slide 21

• Vector Addition Properties:
  1. Commutative property: a + b = b + a.
  2. Associative property: (a + b) + c = a + (b + c).
  3. Additive identity property: a + 0 = a, where 0 is the zero vector.
  4. Additive inverse property: a + (-a) = 0, where -a is the additive inverse of a.

Slide 22

• Scalar Multiplication Properties:
  1. Multiplication by 0: 0a = 0, where 0 is the scalar.
  2. Multiplication by 1: 1a = a, where 1 is the scalar.
  3. Distributive property: c(a + b) = ca + cb, where c is the scalar.
  4. Distributive property: (c + d)a = ca + da, where c and d are scalars.

Slide 23

• Vector Subtraction Properties:
  1. a - b = a + (-b).
  2. a - b = -(b - a).

Slide 24

• Unit Vector Properties:
  1. The magnitude of a unit vector is always 1: |â| = 1.
  2. Any vector a can be expressed as the product of its magnitude and a unit vector: a = |a|â.

Slide 25

• Dot Product Properties:
  1. Commutative property: a · b = b · a.
  2. Distributive property: a · (b + c) = a · b + a · c.
  3. Associative property: a · (b · c) = (a · b) · c.
  4. Scalar multiplication property: c(a · b) = (a · cb), where c is a scalar.

Slide 26

• Angle between Vectors Properties:
  1. If a and b are parallel, the angle between them is 0° or 180°.
  2. If a and b are perpendicular, the angle between them is 90° or π/2 radians.
  3. If a and b are anti-parallel, the angle between them is 0° or 180°.

Slide 27

• Cross Product Properties:
  1. The cross product of two parallel vectors is zero: a × b = 0.
  2. The cross product of two anti-parallel vectors is zero: a × (-b) = 0.
  3. The cross product of two perpendicular vectors is the maximum: |a × b| = |a| * |b|.

Slide 28

• Magnitude and Direction Properties:
  1. The magnitude of a vector a is always positive: |a| ≥ 0.
  2. The magnitude of a vector a is zero if and only if a is the zero vector: |0| = 0.
  3. The direction of a vector a can be specified using its components or unit vector notation.

Slide 29

• Vector Projection Properties:
  1. The scalar projection of a onto b is the length of the projection.
  2. The vector projection of a onto b is the vector formed by the projection.
  3. The projection of a onto b is parallel to b.

Slide 30

• Applications of Vectors:
  1. Force analysis in physics.
  2. Position, displacement, and velocity in kinematics.
  3. Electromagnetic fields and electric circuits.
  4. Optimization and linear programming.
  5. Geometric transformations in computer graphics.