Question

State any four properties of the scalar product of two vectors.

Answer 1

Hint:The scalar product of two vectors is the product of the magnitudes of both the vectors and the cosine of the angle between them. When two vectors are having an angle between them, then their scalar product is in a relationship with the magnitudes of those vectors and the cosine of the angle between them.

Complete step-by-step solution:
Scalar product of two vectors:
The scalar product of two vectors is defined by multiplying their magnitudes with the cosine of the angle between them. The scalar product of orthogonal vectors vanishes and the antiparallel vectors are negative.
Characteristics of Scalar product of two vectors:
The scalar product is commutative.
The two manually perpendicular vectors of a scalar product are zero.
The two parallel and vectors of a scalar product are equal to the product of their magnitudes.
The square of its magnitude is equal to the Self-product of a vector.
Properties of the scalar product of two vectors:
The product quantity \[\overrightarrow A \]\[ \cdot \]\[\overrightarrow B \] is always a scalar. It is positive when the angle between the vectors is acute (i.e.,\[ < {90^ \circ }\]) and negative if the angle between them is obtuse (i.e.\[{90^ \circ } < \theta < {180^ \circ }\]).
The scalar product is commutative.
\[\overrightarrow A .\overrightarrow B = \overrightarrow B .\overrightarrow A .\]
The vectors obey distributive law.
\[\overrightarrow A .\left( {\overrightarrow {B + C} } \right) = \overrightarrow A .\overrightarrow B + \overrightarrow A .\overrightarrow C \]
The angle between the vectors,
\[\theta = {\cos ^{ - 1}}\left[ {\dfrac{{\overrightarrow {A \cdot B} }}{{AB}}} \right]\]

Note:The symbol for the scalar product is the dot$(.)$ and so refers to the scalar product as the dot product.
The combining of two vectors to produce the result is scalar.
Finding the component of a vector in the direction of another vector by using the scalar product.