Vectors - Definition of scalar product of two vectors

• Scalar product of two vectors is a mathematical operation that takes two vectors as inputs and produces a scalar as the result.
• It is also known as dot product or inner product.
• Represented as:
  • For two vectors a and b, the scalar product is denoted as a · b.

Properties of Scalar Product

1. Commutativity: a · b = b · a

2. Distributivity over vector addition: (a + b) · c = a · c + b · c

3. Multiplication by a scalar: (k · a) · b = k · (a · b)

4. Linearity: (k · a) · b = a · (k · b) = k · (a · b)

Scalar Product Formula

• The scalar product of two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is given by: a · b = a₁ * b₁ + a₂ * b₂ + a₃ * b₃
• Note that this formula can be extended to vectors of any dimension.

Geometric Interpretation

• The scalar product of two vectors a and b is positively related to the angle between them.
• If the vectors are parallel, the scalar product is positive.
• If the vectors are perpendicular, the scalar product is zero.
• If the vectors are anti-parallel, the scalar product is negative.

Example 1

Find the scalar product of the vectors a = (2, -3) and b = (4, 5). Solution: a · b = 2 * 4 + (-3) * 5 = 8 - 15 = -7 Therefore, a · b = -7.

Example 2

If the scalar product of two vectors a and b is zero, what does it imply about their angle? Solution: If a · b = 0, it implies that the angle between a and b is 90 degrees (perpendicular).

Equations and Scalar Product

• The scalar product can also be used to derive equations involving vectors.
• For example, the equation a · b = 0 represents a plane that is perpendicular to the vector a.

Application in Physics

• The scalar product has various applications in physics, such as calculating work done, finding component forces, determining angles between vectors, etc.
• It is an essential concept in vector calculus and mechanics.

11. Properties of Scalar Product

• Commutativity: a · b = b · a
• Distributivity over vector addition: (a + b) · c = a · c + b · c
• Multiplication by a scalar: (k · a) · b = k · (a · b)
• Linearity: (k · a) · b = a · (k · b) = k · (a · b)

12. Scalar Product Formula

• The scalar product of two vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is given by: a · b = a₁ * b₁ + a₂ * b₂ + a₃ * b₃

13. Geometric Interpretation

• The scalar product of two vectors a and b is positively related to the angle between them.
• If the vectors are parallel, the scalar product is positive.
• If the vectors are perpendicular, the scalar product is zero.
• If the vectors are anti-parallel, the scalar product is negative.

14. Example 1

• Example: Find the scalar product of the vectors a = (2, -3) and b = (4, 5).
• Solution: a · b = 2 * 4 + (-3) * 5 = 8 - 15 = -7
• Therefore, a · b = -7.

15. Example 2

• Example: If the scalar product of two vectors a and b is zero, what does it imply about their angle?
• Solution: If a · b = 0, it implies that the angle between a and b is 90 degrees (perpendicular).

16. Equations and Scalar Product

• The scalar product can be used to derive equations involving vectors.
• For example, a · b = 0 represents a plane that is perpendicular to the vector a.

17. Application in Physics

• The scalar product has various applications in physics, such as calculating work done, finding component forces, determining angles between vectors, etc.
• It is an essential concept in vector calculus and mechanics.

18. Scalar Product and Vector Projection

• The scalar product can be used to find the projection of one vector onto another.
• The projection of vector a onto vector b is given by: proj_ba = (a · b) / |b|

19. Scalar Product and Vector Projection (Continued)

• The magnitude of the projection is given by |proj_ba| = |a| cos(theta), where theta is the angle between a and b.
• The projection vector is a vector that lies on the same line as b.

20. Geometric Applications of Scalar Product

• The scalar product can be used to determine the angle between two vectors geometrically.
• The angle between vectors a and b is given by: cos(theta) = (a · b) / (|a| |b|)
• The scalar product is also used to find the perpendicular distance from a point to a line.

21. Applications of Scalar Product in Geometry

• The scalar product can be used to determine whether two vectors are perpendicular, parallel, or neither.
• If a · b = 0, then a and b are perpendicular.
• If a · b ≠ 0 and |a · b| = |a| |b|, then a and b are parallel.
• If a · b ≠ 0 and |a · b| ≠ |a| |b|, then a and b are neither parallel nor perpendicular.
• Example: Determine whether the vectors a = (1, 2, 3) and b = (4, -2, 1) are perpendicular, parallel, or neither.

22. Applications of Scalar Product in Geometry (Continued)

• a · b = 1 * 4 + 2 * -2 + 3 * 1 = 4 - 4 + 3 = 3
• |a| = sqrt(1^2 + 2^2 + 3^2) = sqrt(14)
• |b| = sqrt(4^2 + (-2)^2 + 1^2) = sqrt(21)
• |a| |b| = sqrt(14) * sqrt(21) ≈ 10.58
• Since a · b ≠ 0 and |a · b| ≠ |a| |b|, the vectors a and b are neither parallel nor perpendicular.

23. Applications of Scalar Product in Work Done

• The scalar product can be used to calculate work done in physics.
• Work done is given by the formula: Work = Force * Distance * cos(theta)
• The scalar product is used to find the component of force in the direction of displacement.

24. Applications of Scalar Product in Work Done (Continued)

• Example: A force of 30 N is applied to move an object through a displacement of 5 m at an angle of 60 degrees to the direction of the force. Calculate the work done.
• Work = Force * Distance * cos(theta)
• Work = 30 * 5 * cos(60 degrees) = 30 * 5 * 0.5 = 75 J
• Therefore, the work done is 75 Joules.

25. Applications of Scalar Product in Forces

• The scalar product can be used to calculate the component of a force acting in a particular direction.
• Example: A force of 50 N is applied at an angle of 30 degrees to the horizontal. Find the vertical component of the force.

26. Applications of Scalar Product in Forces (Continued)

• Vertical component of the force = Force * cos(theta)
• Vertical component = 50 * cos(30 degrees) ≈ 43.3 N

27. Applications of Scalar Product in Physics

• The scalar product is used in physics to calculate torque.
• Torque = Force * Moment arm * sin(theta), where theta is the angle between the force and the moment arm.

28. Applications of Scalar Product in Physics (Continued)

• Example: A force of 20 N is applied at a moment arm of 3 m at an angle of 45 degrees. Calculate the torque.
• Torque = Force * Moment arm * sin(theta)
• Torque = 20 * 3 * sin(45 degrees) = 20 * 3 * 0.707 ≈ 42.4 Nm

29. Scalar Product in Perpendicular Distance

• The scalar product can be used to find the perpendicular distance from a point to a line.
• Example: Find the perpendicular distance from the point P(1, 2, 3) to the line given by the vector equation r = (2 + t, 1 - 2t, 3t), where t is a parameter.

30. Scalar Product in Perpendicular Distance (Continued)

• The line can be represented as the sum of a position vector and a parallel vector: r = a + tb
• The perpendicular distance from P to the line is given by d = |(OP) · b| / |b|
• Let b = (1, -2, 3) and a = (2, 1, 0)
• (OP) · b = (1 - 2, 2 - 1, 3 - 0) · (1, -2, 3) = (-1 + 4 + 9) = 12
• |b| = sqrt(1^2 + (-2)^2 + 3^2) = sqrt(1 + 4 + 9) = sqrt(14)
• Therefore, d = |(OP) · b| / |b| = 12 / sqrt(14)