Question

Which of the following expressions are meaningful?
A. \[\vec u \cdot (\vec v \times \vec w)\]
B. \[(\vec u \cdot \vec v) \cdot \vec w\]
C. \[(\vec u \cdot \vec v)\vec w\]
D. \[\vec u \times (\vec v \cdot \vec w)\]

Answer 1

Hint: A vector is a measurement with both magnitude and direction. Only a few mathematical operations, such addition and multiplication, can be used on vectors. There are two methods for multiplying vectors: dot product and cross product. Vector algebra includes the idea of the cross product of two vectors. There are several types of vectors, and we may use them for addition, subtraction, and multiplication among other operations.



Formula Used:The scalar triple product of three vectors a, b, and c is
\[\left( {a \times b} \right) \cdot c\]



Complete step by step solution:(A) The expression \[\overrightarrow {\rm{v}} \times \overrightarrow {\rm{w}} \] is a vector and dot product of two vectors is meaningful.
So,\[\overrightarrow {\rm{u}} \cdot (\overrightarrow {\rm{v}} \times \overrightarrow {\rm{w}} )\]is meaningful
Option A's expression make sense because they are the dot product of two vectors.
(B) The term \[\vec u \times \vec v\] is scalar and dot product of scalar.
So,\[(\vec u \times \vec v) \cdot \vec w\]is not meaningful.
Since the expression in choice (b) is the dot product of a vector and a scalar quantity, it has no meaning.
(C) The term\[\overrightarrow {\rm{u}} \times \overrightarrow {\rm{v}} \] is scalar and expression is of the form \[\lambda \overrightarrow {\rm{w}} \]
So, \[(\overrightarrow {\rm{u}} \times \overrightarrow {\rm{v}} ) \cdot \overrightarrow {\rm{w}} \]is meaningful.
Expression in choice (c) makes sense because it multiplies a vector by a scalar.
(D) As\[\vec v \times \vec w\] is scalar and cross product of vector & scalar is not defined.
So, \[\vec u(\vec v \times \vec w)\]is not meaningful.
Option d's expression is useless since it is the cross product of a vector quantity and a scalar quantity.
Therefore, \[\vec u \cdot (\vec v \times \vec w)\]and \[(\vec u \cdot \vec v)\vec w\]are meaningful.



Option ‘A’ and ‘C’ is correct

Note: Students should be very careful in solving cross product and dot product. In three dimensions, the cross product is a binary operation on two vectors. The cross product's final vector is parallel to both initial vectors. The vector components are subjected to a precise process to produce the scalar number known as the dot product.