Question

Which of the following vector identities are false?
A. \[\overrightarrow P + \overrightarrow Q = \overrightarrow Q + \overrightarrow P \]
B. \[\overrightarrow P + \overrightarrow Q = \overrightarrow Q \times \overrightarrow P \]
C. \[\overrightarrow P .\overrightarrow Q = \overrightarrow Q .\overrightarrow P \]
D. \[\overrightarrow P \times \overrightarrow Q \ne \overrightarrow Q \times \overrightarrow P \]

Answer 1

Hint:There are many physical quantities in physics and mathematics which can be grouped into two types that are vectors and scalars. vector quantities are those which have both magnitude and direction, they are represented by an arrow, where length of the arrow shows magnitude and arrow represents direction

Complete step by step answer:
Vector quantities obeys many properties that are:
(i) Commutative property of addition: for any two vectors a and b, \[\overrightarrow a + \overrightarrow b = \overrightarrow b + \overrightarrow a \] (order of addition of two vectors doesn’t matter).

(ii) Associative property of addition: for any three vectors \[\overrightarrow a ,\,\,\overrightarrow b \,\,and\,\,\overrightarrow c \Rightarrow \left( {\overrightarrow a + \overrightarrow b } \right) + \overrightarrow c = \overrightarrow a + \left( {\overrightarrow b + \overrightarrow c } \right)\].

(iii) Additive identity: for any vector \[\overrightarrow a ,\,\, \Rightarrow \overrightarrow a + 0 = 0 + \overrightarrow a = \overrightarrow a \]
(iv) Distributivity of scalar product over addition: let \[\overrightarrow a ,\,\overrightarrow b \,\,and\,\overrightarrow c \] be any three vectors, then \[ \Rightarrow \overrightarrow a .\left( {\overrightarrow b + \overrightarrow c } \right) = \overrightarrow a .\overrightarrow b + \overrightarrow {a.} \overrightarrow c \]

(v) Commutative property of scalar product: for any two vectors \[\overrightarrow a \,\,and\,\,\overrightarrow b \Rightarrow \overrightarrow a .\overrightarrow b = \overrightarrow b .\overrightarrow a \]

(vi) Vector product is not commutative: for any two vectors \[\overrightarrow a \,\,and\,\,\overrightarrow b \]:
\[\Rightarrow \overrightarrow a \times \overrightarrow b = \overline { - b} \times \overrightarrow a \\
\Rightarrow \overrightarrow a \times \overrightarrow b \ne \overrightarrow b \times \overrightarrow a \\ \]
So, going by these properties we can see that option B does not follow the vector property this can be explained as:
\[\overrightarrow P + \overrightarrow Q \] the solution lies in the same plane where \[\overrightarrow P \] and \[\overrightarrow Q \] are located whatever may be the magnitude whereas solution of \[\overrightarrow Q \times \overrightarrow P \] lies perpendicular to the plane where \[\overrightarrow P \] and \[\overrightarrow Q \] are present.Rest all other options follow the properties of vector operation.

Note: To determine the direction of a cross product, we stretch our right hand so that the index finger of the right hand is in the direction of the first vector and the middle finger is in the direction of the second vector. Then, the thumb of the right hand indicates the direction or unit vector $n$.