Vector operations refer to the mathematical operations that are performed on vectors, which are quantities that have both magnitude and direction. This includes addition, subtraction, scalar multiplication, and vector multiplication.
To verify the inequality, you can use the properties of absolute values and the properties of vector operations. First, you can rewrite the inequality as |x||y|<=|x|+|y|. Then, you can use the fact that |xy|=|x||y| and that |x|+|y| is always greater than or equal to |x||y|. Therefore, the inequality holds true.
This inequality is significant because it shows that the magnitude of the product of two vectors is always less than or equal to the sum of the magnitudes of the individual vectors. This property is useful in various applications of vector operations, such as in physics and engineering.
One example of using this inequality in real life is in calculating the maximum force that can be applied in a certain direction without causing a structure to collapse. By representing the forces acting on the structure as vectors and using the inequality |F1F2|<=|F1|+|F2|, we can determine the maximum allowable force for each direction without compromising the stability of the structure.
Some other important properties of vector operations include commutativity (the order of vector addition or multiplication does not affect the result), associativity (the grouping of vector operations does not affect the result), and distributivity (the distribution of scalar multiplication over vector addition or subtraction). These properties allow us to manipulate and simplify vector expressions in various ways.