Vector Addition Operation
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A Vector Addition Operation is a vector operation that is an Addition Operation.
- …
- Counter-Example(s):
- See: Vector Space, Pythagorean Theorem.
References
2014
- http://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction
- Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is :[math]\displaystyle{ \mathbf{a}+\mathbf{b} =(a_1+b_1)\mathbf{e}_1 +(a_2+b_2)\mathbf{e}_2 +(a_3+b_3)\mathbf{e}_3. }[/math] The addition may be represented graphically by placing the tail of the arrow b at the head of the arrow a, and then drawing an arrow from the tail of a to the head of b. The new arrow drawn represents the vector a + b, as illustrated below: This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors that have the same base point, this point will also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c).
The difference of a and b is :[math]\displaystyle{ \mathbf{a}-\mathbf{b} =(a_1-b_1)\mathbf{e}_1 +(a_2-b_2)\mathbf{e}_2 +(a_3-b_3)\mathbf{e}_3. }[/math] Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the tails of a and b at the same point, and then draw an arrow from the head of b to the head of a. This new arrow represents the vector a − b, as illustrated below:
Subtraction of two vectors may also be performed by adding the opposite of the second vector to the first vector, that is, a − b = a + (−b).
- Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is :[math]\displaystyle{ \mathbf{a}+\mathbf{b} =(a_1+b_1)\mathbf{e}_1 +(a_2+b_2)\mathbf{e}_2 +(a_3+b_3)\mathbf{e}_3. }[/math] The addition may be represented graphically by placing the tail of the arrow b at the head of the arrow a, and then drawing an arrow from the tail of a to the head of b. The new arrow drawn represents the vector a + b, as illustrated below: This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors that have the same base point, this point will also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c).
