Vector addition involves adding the components of the vectors together. Geometrically, this can be interpreted as adding the vectors 'tip-to-tail.' In order to add two vectors, they must be in the space with the same number of components (for instance, a two-dimensional vector cannot be added to a three-dimensional vector). For example, let u=[3,1] and v=[2,5]. To add u + v, we must add their respective x and y components together,
In two and three dimensions, vector addition can be interpreted geometrically. The 'tip-to-tail' method involves arranging the vectors to be added together by starting each vector at the end of the previous one (the 'tip' of the first vector is where you place the 'tail' of the next vector.) The end result can be found by finding the vector that starts at the tail of the first vector, and ends at the tip of the last vector. For example, let u=[2,1], v=[4,4], and w=[1,3]. The result is shown below:
The resulting vector u + v + w=[7,8] is the same as is found by adding the components of the three vectors together. Similar addition can be performed with 3 dimensional vectors by placing the vectors tip-to-tail, and finding the vector that runs from the starting point to the end of the last vector to be added. For example, let u=[3,1,-2] and v=[4,0,3]. By adding components, we find that u + v = [3 + 4, 1 + 0, (-2) + 3] = [7,1,1].
(Note that in the above example, u had a negative component in the z direction. Vector components can be any real number, positive or negative.)
When adding vectors in Rn, we can always add them by simply adding their components. If u=[u1, u2, ..., un] and v=[v1, v2, ..., vn], then u + v=[u1 + v1, u2 + v2, ..., un + vn]
1 | Find the sum of the vectors in Rn where n>3Vector subtraction is simply the negative of addition. Let's start in two dimensions for simplicity's sake. Let v=[2,3] and w=[4,1]. Then what is v - w? To find v - w, we use the fact that v - w = v + (-w). Then to find -w, all we have to do is change the sign on all components of w. So since w=[4,1], -w=[-4,-1], and by adding components, v + (-w) =[2 + (-4), 3 + (-1)] = [2-4, 3-1] = [-2, 2]. So v - w=[-2, 2]. Geometrically, taking the negative of a vector makes the vector have the same length, but point in the exact opposite direction. We show the above example geometrically below,
When taking the negative of a vector, it is very important to change the sign of every component of that vector. So, if u=[2, -3, 0, 4] in R4, then -u=[-2, 3, 0, -4]. Note that since the formula for the length of a vector involves squaring each component, then |u| = |-u|. As with vector addition, the vectors involved must all have the same number of components. Geometric vector subtraction in 2 or 3 dimensions is still done by the tip-to-tail method, but the vector to be subtracted must be turned in the opposite direction (which is the equivalent of changing the sign of each component.)
Multiplication involving vectors is not quite the same as ordinary multiplication involving only scalars. Scalar multiplication involves multiplying a scalar number by a vector. Let k be a scalar, and u be a vector in Rn. Since u=[u1, u2, ..., un], then the product ku=[ku1, ku2, ..., kun]. This is scalar multiplication. Each component of the vector u is simply multiplied by the scalar k. The result is a vector, and
If k>0, then ku is in the same direction as u, with length k|u|
If k=0, then ku is the zero vector (the starting point is the same, but the length is zero)
If k<0, then ku is in the opposite direction as u, with length |k||u|
We already used scalar multiplication in our look at subtraction, where (-1)u = -u. As expected, 1u=u. The following table sums up the properties of vector addition/subtraction and scalar multiplication so far.
1) u + v = v + u
2) u + (v + w) = (u + v) + w
3) u - v = u + (-v)
4) k(u + v) = ku + kv
5) Let a,b be scalars. a(bu) = (ab)u = (ba)u
6) (a + b)u = au + bu
7) |ku| = |k||u|
8) u - u = u + (-u) = 0 (The zero vector)
Note an important fact, that |u + v| ≤ |u| + |v|