Vector Algebra and Vector Calculus are used widely in science, especially Physics and engineering.
The physical world involves four dimensions, one scalar dimension and three vector dimensions. From this you can say that 3/4 of the world involve vectors.
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Sure. There are other applications of arithmetic, but algebra without arithmetic is impossible. A broad knowledge of arithmetic is essential for mastery of algebra.
Linear Algebra
Zero is a number (a scalar quantity without unit) while zero vector (or null vector) is a vector quantity having zero magnitude and arbitrary direction.
Use equation.
When focusing upon abstract algebra, there are many different areas included within this topic such as groups, rings, modules and vector space. These all are part of the sequence to constructing abstract algebra.