Select incorrect option
A. for any two vectors $|\overrightarrow A .\overrightarrow B | \leqslant AB$
B. for any two vectors $|\overrightarrow A \times \overrightarrow B | \leqslant AB$
C. a vector is not changed if it is slide parallel to itself
D. a vector is necessarily changed if it is rotated through an angle.

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Hint: We will use the vector dot and cross product formula $|\overrightarrow A .\overrightarrow B | = AB|\cos \theta |$ and $|\overrightarrow A \times \overrightarrow B | = AB|\sin \theta |$ respectively, and then apply that value of sin and cos is always less than or equal to 1. Similarly we check the option C and option D to find the incorrect answer.

Complete step by step answer:
Now from the question, we have
$|\overrightarrow A .\overrightarrow B | = AB|\cos \theta |$
As $\cos \theta \leqslant 1$ so $|\overrightarrow A .\overrightarrow B | \leqslant AB$. So option A. is correct
Now, $|\overrightarrow A \times \overrightarrow B | = AB|\sin \theta |$
As $\sin \theta \leqslant 1$ so $|\overrightarrow A \times \overrightarrow B | \leqslant AB$. So option B. is also correct
When a vector is slid parallel to itself, the vector becomes Acos${0^0}$ = A, it remains the same. So option C. is correct.
When a vector A. is rotated through an angle of ${360^0}$ the vector becomes Acos${360^0}$ = A, it means it remains the same.
Therefore, option D is incorrect
Hence option D. is the correct option.

A vector is an object that has both the direction and therefore the magnitude. The length indicates the magnitude of the vectors, whereas the arrow indicates the direction. There are differing types of vectors. Generally , there are two ways of multiplying vectors.

(i) scalar product of vectors (also referred to as Scalar product)
(ii) vector product of vectors (also referred to as Vector product).
Cross product
The cross product of two vectors a and b is given by a vector whose magnitude is given by |a||b|sin$\theta $(where ${0^0} \leqslant \theta \leqslant {180^0}$) which represents the angle between the 2 vectors and therefore the direction of the resultant vector is given by a unit vector n^ whose direction is perpendicular to both the vectors a and b in such how that a, b and $\mathop n\limits^\^ $ are oriented in right-handed system.

Dot product
The inner product of two vectors a and b of magnitude |a| and |b| is given as |a||b| cos$\theta $, where $\theta $represents the angle between the vectors a and b taken within the direction of the vectors.
We can express the inner product as:
a.b=|a||b| cos$\theta $
where |a| and |b| represent the magnitude of the vectors a and b while cos$\theta $ denotes the cosine of the angle between both the vectors and a.b indicate the scalar product of the 2 vectors.