Which of the following is scalar triple product?
The scalar triple product of three vectors a, b, and c is (a×b)⋅c. It is a scalar product because, just like the dot product, it evaluates to a single number. (In this way, it is unlike the cross product, which is a vector.)
What is meant by scalar triple product?
By the name itself, it is evident that the scalar triple product of vectors means the product of three vectors. It means taking the dot product of one of the vectors with the cross product of the remaining two. It is denoted as. [a b c ] = ( a × b) .
How do you prove a scalar triple product?
To determine the formula for the scalar triple product, the cross product of two vectors is calculated first. After that, the dot product of the remaining vector with the resultant vector is calculated. If the triple product results to be zero, then it suggests that one of the three vectors taken is of zero magnitudes.
Why is a scalar triple product zero?
If any two of three vectors are equal or parallel, then the scalar triple product is zero. If three vectors are coplanar then \[→a\]\[→b\]\[→c\]abc=0.
What are the properties of scalar product?
Properties of the scalar product
- The scalar product of a vector and itself is a positive real number: u → ⋅ u → ⩾ 0 .
- The scalar product is commutative: u → ⋅ v → = v → ⋅ u → .
- The scalar product is pseudoassociative: α ( u → ⋅ v → ) = ( α u → ) ⋅ v → = u → ⋅ ( α v → ) where is a real number.
What is AXB XC?
(a x b) x c = (a c)b – (b c)a (1) for the repeated vector cross product. This vector-valued identity is easily seen to be. completely equivalent to the scalar-valued identity.
What does Vector triple product represent?
The “vector triple product” is nothing more than the determinant of a matrix. In particular, if you take the columns of a matrix to be your three vectors, the the determinant of the matrix is equal to the triple product.
What is BAC cab rule?
linear-algebra vectors. These are examples of BAC-CAB rule in a physics book.( →A×→B)⋅(→C×→D)=(→A⋅→C)(→B⋅→D)−(→A⋅→D)(→B⋅→C) →A×(→B×(→C×→D))=→B(→A⋅(→C×→D))−(→A⋅→B)(→C⋅→D)
Which is the vector triple product?
The cross-product of the vectors such as a × (b × c) and (a × b) × c is known as the vector triple product of a, b, c. The vector triple product a × (b × c) is a linear combination of those two vectors which are within brackets. The ‘r’ vector r=a×(b×c) is perpendicular to a vector and remains in the b and c plane.
What are the six properties of scalar product?
Properties of scalar product of two vectors are:
- (1) The product quantity→A . →B is always a scalar.
- (2) The scalar product is commutative, i.e. →A →B ≠ →B . →A.
- (3) The vectors obey distributive law i.e →A (→B + →C ) = →A . →B + →A .
- (4) The angle between the vectors θ = cos-1 [→A. →BAB
What property is ax BxC AxB xC?
Associative property of multiplication
Mathematics Glossary » Table 3
| Associative property of addition | (a +b) + c = a + (b+c) |
|---|---|
| Associative property of multiplication | (a x b) x c = a x (b x c) |
| Commutative property of multplication | a x b = b x a |
| Multiplicative identity property 1 | a x 1 = 1 x a = a |
What is scalar triple product and vector triple product?
Students can go through this article to learn more about the scalar triple product and vector triple product, its definition, formula, properties and more. Scalar triple product formula means the dot product of one of the vectors with the cross product of the other two vectors. It can be written as:
How do you solve vector triple product problems?
Let’s solve some vector triple product example problems. There are three vectors known as →a, →b, and→c. The magnitudes of these vectors are | →a | = 1, | →b | = 2, | →c | = 1. The equation is →a × (→a × →b) + →c=0.
How do you find the volume of a parallelepiped using scalar triple product?
The dot product of the resultant with c will only be zero if the vector c also lies in the same plane. This is because the angle between the resultant and C will be and cos . Thus, by the use of the scalar triple product, we can easily find out the volume of a given parallelepiped.
What is the product of three vectors?
The product of three vectors in mathematics simply refers to the scalar triple product of vectors. The resultant vector is a scalar quantity and is represented as (a x b).c. In this formula, dot and cross can be interchanged, that is; (a x b).c = a. (b x c).