Vectors - Geometric Interpretation of STP

  • In this lecture, we will discuss the geometric interpretation of the scalar triple product (STP) of vectors
  • STP is a scalar quantity that determines the volume of the parallelipiped formed by three vectors
  • It is denoted by [a, b, c], where a, b, and c are vectors

STP and the Volume of a Parallelipiped

  • The scalar triple product [a, b, c] is given by the dot product of the first vector with the cross product of the other two vectors: [a, b, c] = a · (b × c)
  • The absolute value of the STP represents the volume of the parallelipiped formed by the three vectors

Geometrical Significance of STP

  • If the value of the STP is zero, it means that the three vectors are coplanar (lie on the same plane)
  • If the value of the STP is positive, it means that the three vectors form a right-handed system and the parallelipiped is oriented in the positive direction
  • If the value of the STP is negative, it means that the three vectors form a left-handed system and the parallelipiped is oriented in the negative direction

Example 1

Let’s consider three vectors:

  • a = 2i + 3j - 4k
  • b = i - 2j + k
  • c = 3i - j + 5k The STP of these vectors can be calculated as follows: [a, b, c] = a · (b × c)

Example 1 (contd.)

To find the cross product (b × c): b × c = (i - 2j + k) × (3i - j + 5k) Using the determinant method, we can calculate the cross product as follows: | i j k | | 1 -2 1 | | 3 -1 5 |

Example 1 (contd.)

Using the determinant method, we can calculate the cross product as follows: b × c = (10i + 6j - 5k) - (3i - j + 5k) + (6i + 2j + k) = 13i + 7j + i - 3k = 14i + 7j - 3k

Example 1 (contd.)

Now, let’s calculate the dot product of vector a with the cross product (b × c): a · (b × c) = (2i + 3j - 4k) · (14i + 7j - 3k) = 28 + 21 - 12 = 37 Therefore, [a, b, c] = 37

Geometrical Interpretation of Example 1

  • Since [a, b, c] = 37, the volume of the parallelipiped formed by vectors a, b, and c is 37 cubic units
  • The orientation of the parallelipiped can be determined by the sign of the STP, which in this case is positive, indicating a right-handed system
  1. Geometrical Interpretation of Example 1 (contd.)
  • Since [a, b, c] = 37, the volume of the parallelipiped formed by vectors a, b, and c is 37 cubic units
  • The orientation of the parallelipiped can be determined by the sign of the STP, which in this case is positive, indicating a right-handed system
  1. Example 2 Consider the following vectors:
  • a = i + 2j - k
  • b = 2i - j + 3k
  • c = 3i + j - 2k Let’s calculate the STP of these vectors and interpret the results.
  1. Example 2 (contd.) To find the cross product (b × c): b × c = (2i - j + 3k) × (3i + j - 2k) Using the determinant method, we can calculate the cross product as follows: | i j k | | 2 -1 3 | | 3 1 -2 |
  1. Example 2 (contd.) Using the determinant method, we can calculate the cross product as follows: b × c = (6i + 2j - 3k) - (3i - 2k + 3i) + (2j + 3k + 3i) = 0i + 0j + 8k = 8k
  1. Example 2 (contd.) Now, let’s calculate the dot product of vector a with the cross product (b × c): a · (b × c) = (i + 2j - k) · 8k = -8 Therefore, [a, b, c] = -8
  1. Geometrical Interpretation of Example 2
  • Since [a, b, c] = -8, the volume of the parallelipiped formed by vectors a, b, and c is -8 cubic units
  • The negative sign indicates a left-handed system
  1. Example 3 Consider the following vectors:
  • a = i + j + k
  • b = -i + j - k
  • c = i - j + k Let’s calculate the STP of these vectors and interpret the results.
  1. Example 3 (contd.) To find the cross product (b × c): b × c = (-i + j - k) × (i - j + k) Using the determinant method, we can calculate the cross product as follows: | i j k | | -1 1 -1 | | 1 -1 1 |
  1. Example 3 (contd.) Using the determinant method, we can calculate the cross product as follows: b × c = (0i + 2j + 0k) - (-i - k + j) + (i - j - k) = 2i + 4j + 4k
  1. Example 3 (contd.) Now, let’s calculate the dot product of vector a with the cross product (b × c): a · (b × c) = (i + j + k) · (2i + 4j + 4k) = 2 + 4 + 4 = 10 Therefore, [a, b, c] = 10
  1. Geometrical Interpretation of Example 3 (contd.)
  • Since [a, b, c] = 10, the volume of the parallelipiped formed by vectors a, b, and c is 10 cubic units
  • The positive sign indicates a right-handed system
  1. Example 4 Consider the following vectors:
  • a = 3i + j - k
  • b = i - 2j + k
  • c = 2i + 3j - 4k Let’s calculate the STP of these vectors and interpret the results.
  1. Example 4 (contd.) To find the cross product (b × c): b × c = (i - 2j + k) × (2i + 3j - 4k) Using the determinant method, we can calculate the cross product as follows: | i j k | | 1 -2 1 | | 2 3 -4 |
  1. Example 4 (contd.) Using the determinant method, we can calculate the cross product as follows: b × c = (-2i - 4j - 4k) - 3i + 8k + 2j = -5i - 2j + 4k
  1. Example 4 (contd.) Now, let’s calculate the dot product of vector a with the cross product (b × c): a · (b × c) = (3i + j - k) · (-5i - 2j + 4k) = -15 - 2 - 4 = -21 Therefore, [a, b, c] = -21
  1. Geometrical Interpretation of Example 4
  • Since [a, b, c] = -21, the volume of the parallelipiped formed by vectors a, b, and c is -21 cubic units
  • The negative sign indicates a left-handed system
  1. Properties of STP
  • STP is a scalar quantity
  • The STP of three vectors is invariant under cyclic interchange, i.e., [a, b, c] = [b, c, a] = [c, a, b]
  • The STP of three vectors changes sign under anticyclic interchange, i.e., [a, b, c] = -[c, b, a] = -[b, a, c]
  1. Properties of STP (contd.)
  • The STP of three vectors is proportional to the volume of the parallelipiped formed by the vectors: [ka, kb, kc] = (k^3)[a, b, c], where k is a scalar
  • The STP of three collinear vectors is zero, as the volume of the parallelipiped becomes zero
  1. Applications of STP
  • The STP is used in various fields such as physics, engineering, and computer science for solving problems involving the calculation of volume, area, and orientation of objects
  • It is also used in geometry for determining the non-collinearity of vectors and for proving geometric theorems
  1. Summary
  • The scalar triple product (STP) of three vectors, denoted as [a, b, c], represents the volume of the parallelipiped formed by the vectors
  • The STP has a geometrical interpretation, where the sign indicates the orientation of the parallelipiped
  • We can calculate the STP by taking the dot product of one vector with the cross product of the other two vectors
  • The properties of STP include invariance under cyclic interchange and sign change under anticyclic interchange