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The sum of two forces F 1 and F 2 applied at A can be computed from the sum of the segments that define them. Let F 1 = B−A and F 2 = D−A, then the sum of these two vectors is = + = +, which can be written as
When V is Euclidean n-space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in Euclidean space. In this sense, the unit dyadic ij is the function from 3-space to itself sending a 1 i + a 2 j + a 3 k to a 2 i, and jj sends this sum to a 2 j.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
The binary operation, called vector addition or simply addition assigns to any two vectors v and w in V a third vector in V which is commonly written as v + w, and called the sum of these two vectors. The binary function, called scalar multiplication, assigns to any scalar a in F and any vector v in V another vector in V, which is denoted av ...
Minkowski sums act linearly on the perimeter of two-dimensional convex bodies: the perimeter of the sum equals the sum of perimeters. Additionally, if K {\textstyle K} is (the interior of) a curve of constant width , then the Minkowski sum of K {\textstyle K} and of its 180° rotation is a disk.
The addition of two vectors a and b. 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.