Which of the following are true about the triple scalar product of three vectors A, B, C?
Understand the Problem
The question is asking about the properties of the triple scalar product of three vectors A, B, and C. It requires the identification of true statements regarding the relationship between these vectors based on the given options.
Answer
The true statements are the first and second: $A \cdot (B \times C)$ represents the volume, and $A \cdot (B \times C) = C \cdot (A \cdot B)$.
Answer for screen readers
The true statements are:
- A.(BXC) gives volume of parallelepiped formed by vectors A, B, C.
- A.(BXC) = CX(A.B).
Steps to Solve
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Understanding Triple Scalar Product
The triple scalar product of three vectors $A$, $B$, and $C$ is given as $A \cdot (B \times C)$. This represents the volume of the parallelepiped formed by the vectors. -
Evaluating Statement A
The statement "A.(BXC) gives volume of parallelepiped formed by vectors A, B, C" is true. The expression $A \cdot (B \times C)$ indeed calculates the volume. -
Evaluating Statement B
The statement "A.(BXC) = CX(A.B)" needs to be checked. This can also be rearranged to $A \cdot (B \times C) = C \cdot (A \times B)$, which is equal because of the cyclic property of scalar triple products. Therefore, this statement is true. -
Evaluating Statement C
The statement "A.(BXC) = B.(CXA)" also needs to be evaluated. By the cyclic property of scalar products, we have $A \cdot (B \times C) = B \cdot (C \times A)$, thus the equality does not hold. This statement is false. -
Evaluating Statement D
Statement "A.(BXC) = -B.(AXC)" is to be checked. This can be rewritten as $A \cdot (B \times C) = -B \cdot (A \times C)$. However, both sides cannot be equated directly; hence this is also false.
The true statements are:
- A.(BXC) gives volume of parallelepiped formed by vectors A, B, C.
- A.(BXC) = CX(A.B).
More Information
The triple scalar product effectively captures the geometric interpretation of volume in three-dimensional space created by the three vectors. It also exhibits properties such as cyclic permutation and anti-symmetry, which are useful in vector calculus.
Tips
- Confusing the ordering of vectors in scalar products can result in incorrect evaluations. Always maintain the correct sequence, especially in triple products.
- Misapplying properties such as the cyclic property; remembering that only the scalar value switches when cyclically permuting vectors is key.
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