Vectors - Identifying Coplanar Vectors using STP
- In this lesson, we will learn about coplanar vectors and how to identify them using the scalar triple product (STP).
- Coplanar vectors are vectors that lie in the same plane.
- The scalar triple product is a method used to determine whether three given vectors are coplanar or not.
- Let’s understand this concept through examples and equations.
- Reminder: Do not include any comments, especially at the start or end of your responses.
Definition of Coplanar Vectors
- Coplanar vectors are vectors that lie on the same plane.
- When three vectors lie on the same plane, they are said to be coplanar.
- This can be understood visually if the vectors lie within a single plane.
- Let’s move on to how we can identify coplanar vectors using the scalar triple product.
Scalar Triple Product (STP)
- The scalar triple product (STP) is a mathematical expression used to determine whether three vectors are coplanar or not.
- The scalar triple product of three vectors A, B, and C is given by the formula:
- STP(A, B, C) = A · (B × C)
- Here, “·” represents the dot product, and “×” represents the cross product of vectors.
- If the scalar triple product of three vectors is zero, then the vectors are coplanar.
- If the scalar triple product is non-zero, then the vectors are not coplanar.
Example 1
- Let’s consider three vectors: A = <2, 3, -1>, B = <4, 1, 5>, C = <6, 7, 8>.
- To determine whether these vectors are coplanar or not, we will calculate the scalar triple product using the formula: STP(A, B, C) = A · (B × C).
- A · (B × C) = <2, 3, -1> · ((4, 1, 5) × (6, 7, 8))
- Calculating the cross product: (B × C) = <18, 14, -22> - <30, -26, -8> = <-12, -12, -30>
- A · (B × C) = <2, 3, -1> · <-12, -12, -30>
- Dot product calculation: A · (B × C) = (2 * -12) + (3 * -12) + (-1 * -30) = -24 - 36 + 30
- A · (B × C) = -30
- Since the scalar triple product is not zero, these vectors are not coplanar.
Example 2
- Let’s consider three vectors: A = <1, 2, 3>, B = <4, 5, 6>, C = <7, 8, 9>.
- We will calculate the scalar triple product using the formula: STP(A, B, C) = A · (B × C).
- A · (B × C) = <1, 2, 3> · ((4, 5, 6) × (7, 8, 9))
- Calculating the cross product: (B × C) = <18, -14, 6> - <27, -12, -18> = <-9, -26, 24>
- A · (B × C) = <1, 2, 3> · <-9, -26, 24>
- Dot product calculation: A · (B × C) = (1 * -9) + (2 * -26) + (3 * 24) = -9 - 52 + 72
- A · (B × C) = 11
- Since the scalar triple product is not zero, these vectors are not coplanar.
Conclusion
- The scalar triple product (STP) is a valuable tool in determining whether three vectors are coplanar or not.
- If the scalar triple product of three vectors is zero, the vectors are coplanar.
- If the scalar triple product is non-zero, the vectors are not coplanar.
- It is essential to understand and apply this concept when dealing with vectors and their properties.
Slide 11
- Coplanar vectors lie in the same plane.
- The scalar triple product (STP) is used to determine coplanarity.
- STP(A, B, C) = A · (B × C)
- If STP(A, B, C) = 0, then the vectors are coplanar.
- If STP(A, B, C) ≠ 0, then the vectors are not coplanar.
Slide 12
- Let’s consider three vectors: A = <1, 2, 3>, B = <4, 5, 6>, C = <7, 8, 9>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
- (B × C) = <18, -14, 6> - <27, -12, -18> = <-9, -26, 24>
- A · (B × C) = <1, 2, 3> · <-9, -26, 24>
Slide 13
- Dot product calculation: A · (B × C) = (1 * -9) + (2 * -26) + (3 * 24) = -9 - 52 + 72
- A · (B × C) = 11
- Since the scalar triple product is not zero, these vectors are not coplanar.
Slide 14
- Let’s consider three vectors: A = <2, 3, -1>, B = <4, 1, 5>, C = <6, 7, 8>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
- (B × C) = <18, 14, -22> - <30, -26, -8> = <-12, -12, -30>
- A · (B × C) = <2, 3, -1> · <-12, -12, -30>
Slide 15
- Dot product calculation: A · (B × C) = (2 * -12) + (3 * -12) + (-1 * -30) = -24 - 36 + 30
- A · (B × C) = -30
- Since the scalar triple product is not zero, these vectors are not coplanar.
Slide 16
- Consider vectors A = <1, 2, -1>, B = <3, 4, -2>, C = <5, 6, -3>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
Slide 17
- (B × C) = <-6, 12, -6> - <8, -16, 8> = <-14, 28, -14>
- A · (B × C) = <1, 2, -1> · <-14, 28, -14>
Slide 18
- Dot product calculation: A · (B × C) = (1 * -14) + (2 * 28) + (-1 * -14) = -14 + 56 + 14
- A · (B × C) = 56
- Since the scalar triple product is not zero, these vectors are not coplanar.
Slide 19
- Consider vectors A = <2, -1, 3>, B = <4, -2, 6>, C = <6, -3, 9>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
Slide 20
- (B × C) = <-6, 12, -6> - <-6, 12, -6> = <0, 0, 0>
- A · (B × C) = <2, -1, 3> · <0, 0, 0>
- Dot product calculation: A · (B × C) = 0
- Since the scalar triple product is zero, these vectors are coplanar.
Slide 21
- Coplanar vectors lie in the same plane.
- The scalar triple product (STP) is used to determine coplanarity.
- STP(A, B, C) = A · (B × C)
- If STP(A, B, C) = 0, then the vectors are coplanar.
- If STP(A, B, C) ≠ 0, then the vectors are not coplanar.
Slide 22
- Let’s consider three vectors: A = <1, 2, 3>, B = <4, 5, 6>, C = <7, 8, 9>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
- (B × C) = <18, -14, 6> - <27, -12, -18> = <-9, -26, 24>
- A · (B × C) = <1, 2, 3> · <-9, -26, 24>
Slide 23
- Dot product calculation: A · (B × C) = (1 * -9) + (2 * -26) + (3 * 24) = -9 - 52 + 72
- A · (B × C) = 11
- Since the scalar triple product is not zero, these vectors are not coplanar.
Slide 24
- Let’s consider three vectors: A = <2, 3, -1>, B = <4, 1, 5>, C = <6, 7, 8>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
- (B × C) = <18, 14, -22> - <30, -26, -8> = <-12, -12, -30>
- A · (B × C) = <2, 3, -1> · <-12, -12, -30>
Slide 25
- Dot product calculation: A · (B × C) = (2 * -12) + (3 * -12) + (-1 * -30) = -24 - 36 + 30
- A · (B × C) = -30
- Since the scalar triple product is not zero, these vectors are not coplanar.
Slide 26
- Consider vectors A = <1, 2, -1>, B = <3, 4, -2>, C = <5, 6, -3>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
Slide 27
- (B × C) = <-6, 12, -6> - <8, -16, 8> = <-14, 28, -14>
- A · (B × C) = <1, 2, -1> · <-14, 28, -14>
Slide 28
- Dot product calculation: A · (B × C) = (1 * -14) + (2 * 28) + (-1 * -14) = -14 + 56 + 14
- A · (B × C) = 56
- Since the scalar triple product is not zero, these vectors are not coplanar.
Slide 29
- Consider vectors A = <2, -1, 3>, B = <4, -2, 6>, C = <6, -3, 9>.
- Calculate the scalar triple product using STP(A, B, C) = A · (B × C).
Slide 30
- (B × C) = <-6, 12, -6> - <-6, 12, -6> = <0, 0, 0>
- A · (B × C) = <2, -1, 3> · <0, 0, 0>
- Dot product calculation: A · (B × C) = 0
- Since the scalar triple product is zero, these vectors are coplanar.