Vectors - Scalar Triple Product and Handed Systems

  • The scalar triple product is a mathematical operation involving three vectors in three-dimensional space.
  • It is used to determine if three vectors are coplanar or not.
  • The scalar triple product is denoted by [abc].
  • It is defined as the dot product of one vector with the cross product of the other two vectors.
  • The scalar triple product can be calculated as follows: [abc] = a · (b × c), where a, b, and c are vectors. Example: Given the vectors a = (2, 3, 4), b = (-1, 2, 3), and c = (0, 1, -2), find the scalar triple product [abc]. Solution:
  1. Calculate the cross product of b and c: b × c = ((2 * -2) - (3 * 1), (3 * 0) - (-1 * -2), (-1 * 1) - (2 * 0)) = (-7, 2, -1)
  1. Calculate the dot product of a and the result from step 1: a · (b × c) = (2 * -7) + (3 * 2) + (4 * -1) = -14 + 6 - 4 = -12
  1. Therefore, the scalar triple product [abc] = -12.

Handed Systems

  • In mathematics, a handed system refers to the orientation of three vectors in three-dimensional space.
  • There are two possible handed systems: right-handed and left-handed.
  • A right-handed system is one where the vectors follow the right-hand rule, while a left-handed system follows the left-hand rule.
  • The right-hand rule states that if the fingers of the right hand curl in the direction of rotation from the first vector to the second vector, then the thumb points in the direction of the cross product of the two vectors.
  • The left-hand rule is the opposite of the right-hand rule. Example: Given vectors a = (2, 0, 0) and b = (0, 3, 0), determine if the handed system formed by these vectors is right-handed or left-handed. Solution:
  1. Calculate the cross product of a and b: a × b = ((0 * 0) - (0 * 3), (0 * 0) - (2 * 0), (2 * 3) - (0 * 0)) = (0, 0, 6)
  1. Use the right-hand rule to determine the orientation:
    • Point your right hand fingers in the direction of a.
    • Curl your fingers towards b.
    • Your thumb should point in the direction of the cross product (0, 0, 6).
  1. Since the thumb points in the positive z-direction, the handed system formed by vectors a and b is right-handed.

Scalar Triple Product

  • The scalar triple product [abc] can be calculated as the determinant of a 3x3 matrix containing the components of vectors a, b, and c.
  • The determinant of a 3x3 matrix is calculated using the formula: [abc] = a1(b2c3 - b3c2) - a2(b1c3 - b3c1) + a3(b1c2 - b2c1)
  • The scalar triple product is a scalar value and can be positive, negative, or zero. Example: Given vectors a = (2, 1, -3), b = (-1, 0, 2), and c = (4, -1, 3), find the scalar triple product [abc]. Solution:
  1. Use the determinant formula to calculate [abc]: [abc] = 2(0 * 3 - 2 * -1) - 1(-1 * 3 - 4 * 2) - 3(-1 * -1 - 4 * 0) = 2(0 + 2) - 1(-3 - 8) - 3(1 - 0) = 4 - 11 - 3 = -10
  1. Therefore, the scalar triple product [abc] = -10.

Coplanarity of Vectors

  • If the scalar triple product [abc] is equal to zero, the three vectors a, b, and c are coplanar.
  • Coplanar vectors lie on the same plane in three-dimensional space.
  • Coplanar vectors can be represented by a linear combination of two other vectors in the plane.
  • If [abc] is not equal to zero, the three vectors are not coplanar. Example: Given vectors a = (1, 2, 0), b = (-1, 3, -2), and c = (2, -1, 4), determine if they are coplanar. Solution:
  1. Calculate the scalar triple product [abc]: [abc] = 1(3 * 4 - -2 * -1) - 2((-1) * 4 - 2 * 2) + 0((-1) * (-1) - 3 * 2) = 1(12 - 2) - 2(-4 - 4) + 0(1 - 6) = 10 - 16 + 0 = -6
  1. Since [abc] is not equal to zero, the vectors a, b, and c are not coplanar.

Right-Handed Systems

  • In a right-handed system, if you use your right hand to point your index finger in the direction of the first vector and your middle finger in the direction of the second vector, then your thumb will point in the direction of the cross product.
  • The right-hand rule is used to determine the direction of the cross product in a three-dimensional space. Example: Given vectors a = (2, 0, 0) and b = (0, 3, 0), find the cross product a × b and determine its direction using the right-hand rule. Solution:
  1. Calculate the cross product of a and b: a × b = ((0 * 0) - (0 * 3), (0 * 0) - (2 * 0), (2 * 3) - (0 * 0)) = (0, 0, 6)
  1. Use the right-hand rule:
    • Point your right hand index finger in the direction of a (positive x-axis).
    • Curl your right hand middle finger towards b (positive y-axis).
    • Your right hand thumb should point in the direction of the cross product (positive z-axis).
  1. Therefore, the cross product a × b has a direction along the positive z-axis.

Left-Handed Systems

  • In a left-handed system, if you use your left hand to point your index finger in the direction of the first vector and your middle finger in the direction of the second vector, then your thumb will point in the opposite direction of the cross product.
  • The left-hand rule is the opposite of the right-hand rule. Example: Given vectors a = (1, 0, 0) and b = (0, 2, 0), find the cross product a × b and determine its direction using the left-hand rule. Solution:
  1. Calculate the cross product of a and b: a × b = ((0 * 0) - (0 * 2), (0 * 0) - (1 * 0), (1 * 2) - (0 * 0)) = (0, 0, 2)
  1. Use the left-hand rule:
    • Point your left hand index finger in the direction of a (positive x-axis).
    • Curl your left hand middle finger towards b (positive y-axis).
    • Your left hand thumb should point in the opposite direction of the cross product (negative z-axis).
  1. Therefore, the cross product a × b has a direction along the negative z-axis.

Properties of the Scalar Triple Product

  • The scalar triple product satisfies the following properties:
    1. [abc] = -[acb]: Changing the order of the vectors negates the scalar triple product.
    2. [abc] = -[bca] = -[cab]: Cyclic permutation of the vectors negates the scalar triple product.
    3. [abc] = [cba]: Changing the order of the vectors does not change the scalar triple product. Example: Given vectors a = (1, 2, -3), b = (2, -1, 2), and c = (-1, 0, 1), verify the properties of the scalar triple product. Solution:
  1. Calculate [abc]: [abc] = 1((-1) * 1 - 2 * 0) - 2(2 * 1 - (-1) * (-1)) - (-3)(2 * 0 - (-1) * (-1)) = 1(-1) - 2(2 + 1) + 3(0 + 1) = -1 - 2(3) + 3(1) = -1 - 6 + 3 = -4
  1. Verify the properties:
    • [abc] = -[acb]: -4 = -[acb] → [acb] = 4
    • [abc] = -[bca]: -4 = -[bca] → [bca] = 4
    • [abc] = [cba]: -4 = [cba]
  1. Therefore, the scalar triple product satisfies the properties.

Application: Volume Calculations

  • The scalar triple product can be used to calculate the volume of a parallelepiped formed by three vectors.
  • The absolute value of the scalar triple product gives the magnitude of the volume.
  • The volume can also be expressed as the scalar triple product divided by the product of the magnitudes of the vectors. Example: Given vectors a = (1, 2, -1), b = (-2, -1, 3), and c = (3, -1, 2), find the volume of the parallelepiped formed by these vectors. Solution:
  1. Calculate the scalar triple product [abc]: [abc] = 1((-1) * 2 - 3 * (-1)) - 2((-2) * 2 - 3 * 3) - (-1)((-2) * (-1) - (-1) * 3) = 1(-4 + 3) - 2(-4 - 9) + 3(2 + 3) = 1(-1) - 2(-13) + 3(5) = -1 + 26 + 15 = 40
  1. Calculate the magnitudes of the vectors: |a| = √(1^2 + 2^2 + (-1)^2) = √6 |b| = √((-2)^2 + (-1)^2 + 3^2) = √14 |c| = √(3^2 + (-1)^2 + 2^2) = √14
  1. Calculate the volume: Volume = |[abc]| / (|a| * |b| * |c|) = |40| / (√6 * √14 * √14) = 40 / (√6 * 14) = 40 / (√84)
  1. Therefore, the volume of the parallelepiped formed by the vectors a, b, and c is 40 / (√84).

Application: Determining Collinearity of Vectors

  • The scalar triple product can also be used to determine if three vectors are collinear (lie on the same line).
  • If the scalar triple product [abc] is zero, then the vectors a, b, and c are collinear.
  • Collinear vectors can be represented as a linear combination of the other two vectors. Example: Given vectors a = (2, -1, 1), b = (4, -2, 2), and c = (6, -3, 3), determine if they are collinear. Solution:
  1. Calculate the scalar triple product [abc]: [abc] = 2((-2) * 3 - 2 * (-3)) - (-1)(4 * 3 - 2 * 6) + 1(4 * (-3) - (-2) * 6) = 2(-6 + 6) - (-1)(12 - 12) + 1((-12) - (-12)) = 2(0) - (-1)(0) + 1(0) = 0
  1. Since [abc] is equal to zero, the vectors a, b, and c are collinear.

Properties of Collinear Vectors

  • If three vectors a, b, and c are collinear, then they are proportional to each other.
  • Collinear vectors can be written as k times another vector u, where k is a scalar. Example: Given vectors a = (2, -1, 1), b = (4, -2, 2), and c = (6, -3, 3), determine if they are proportional to each other. Solution:
  1. Calculate the magnitudes of the vectors: |a| = √(2^2 + (-1)^2 + 1^2) = √6 |b| = √(4^2 + (-2)^2 + 2^2) = √24 |c| = √(6^2 + (-3)^2 + 3^2) = √54
  1. Determine the scalar k: k = |a| / |b| = √6 / √24 = √(6 / 24) = √(1 / 4) = 1 / 2
  1. Check if b = ka: b = (4, -2, 2) ka = (1/2)(2, -1, 1) = (1, -1/2, 1/2)
  1. Compare b with ka: b = ka (4, -2, 2) = (1, -1/2, 1/2)
  1. Since b is equal to ka, vector b is proportional to vector a.
  1. Repeat the same steps for vector c to check its proportionality with vectors a and b.
  1. Therefore, vectors a, b, and c are proportional to each other.

Applications: Physics and Engineering

  • The concepts of vector operations, scalar triple product, and handed systems have various applications in physics and engineering.
  • Calculation of torque: The cross product of a force vector and a position vector can give the torque or moment of a force.
  • Analysis of forces and moments: By understanding the handedness of a system, engineers can determine the direction of forces and moments acting on an object.
  • Electromagnetism: The right-hand rule is used to determine the direction of magnetic fields and electric currents.
  • Robotics and kinematics: Vector operations are essential in determining the position, velocity, and acceleration of robotic systems. Example: In robotics, the concept of vectors and handed systems are crucial for determining the position and orientation of robot arms. By using the right-hand rule, engineers can determine the direction of joint axes and the rotation of the robot arm.

Summary

  • The scalar triple product [abc] can be calculated as the dot product of one vector with the cross product of the other two vectors.
  • It is used to determine if three vectors are coplanar or not.
  • The right-hand rule is used to determine the direction of the cross product in a right-handed system.
  • The left-hand rule is the opposite of the right-hand rule.
  • The scalar triple product has properties such as negation with changing the order of vectors and cyclic permutation.
  • It can be used to calculate the volume of a parallelepiped and determine the collinearity of vectors.
  • The concepts of vector operations and handed systems have applications in physics, engineering, electromagnetism, and robotics.

Slide 21

  • The scalar triple product can be used to find the angle between two vectors.
  • The angle θ between two vectors a and b can be calculated using the formula: cos(θ) = [ab] / (|a| * |b|)
  • The angle can also be found using the dot product of the normalized vectors as: cos(θ) = a · b / (|a| * |b|) Example: Given vectors a = (3, 4) and b = (-1, 2), find the angle between them. Solution:
  1. Calculate the dot product of the vectors a and b: a · b = (3 * -1) + (4 * 2) = -3 + 8 = 5
  1. Calculate the magnitudes of the vectors: |a| = √(3^2 + 4^2) = √25 = 5 |b| = √((-1)^2 + 2^2) = √5
  1. Use the formula to find the angle: cos(θ) = a · b / (|a| * |b|) = 5 / (5 * √5) = 1 / √5 θ = cos^(-1)(1 / √5) ≈ 26.57 degrees
  1. Therefore, the angle between vectors a and b is approximately 26.57 degrees.

Slide 22

  • The scalar triple product can also be used to find the area of a triangle formed by two vectors.
  • The area of a triangle can be calculated using the formula: Area = 1/2 * |(a × b)|
  • The magnitude of the cross product of the vectors a and b gives the area of the triangle. Example: Given vectors a = (1, 2) and b = (-3, 4), find the area of the triangle formed by these vectors. Solution:
  1. Calculate the cross product of the vectors a and b: a × b = ((1 * 4) - (2 * -3)) = 10