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Home / Linear Algebra / Vectors and Scalar

Vectors and Scalar

Consider two vectors u and v in Rn say
u=(a1, a2 , a3,…………….., an) and v= (b1, b2 , b3,……………..bn)
Their sum, written u + v, is the vector obtained by adding corresponding components from u and v. That is,
u + v = (a1 + b1, a2 + b2, a3 + b3,…………… an + bn)
The scalar product or, simply, product of their vector u by a real number k, written ku, is the vector obtained by multiplying each component of u by k. That is,
ku = k(a1, a2 , a3,…………….., an) = (k a1, k a2 , k a3,…………….., k an)

Dot (inner) product

Consider arbitrary vectors u and v in Rn say
u=(a1, a2 , a3,…………….., an) and v= (b1, b2 , b3,……………..bn)
The dot product or inner product or scalar product of u and v is denoted and defined by
u.v = a1 b1+ a2 b2+ —————— + an bn
That is u.v is obtained multiplying corresponding components and adding resulting products.
Orthogonal or perpendicular: The vector u and v are said to be orthogonal (or perpendicular) if their dot product is zero, that is, if u.v =0.
Example: u=(1, -2, 3) and v= (2, 7, 4) then,
u.v = 1.2-2.7+3.4=2-14+12
Norm (length) of a vector: The norm or length of a vector u in Rn, denoted ||u||, is defined to be the nonnegative square root of u.u. In particular, if
u=(a1, a2 , a3,…………….., an), then
||u|| = √(a21+a22 +a23+……………..+a2 n)
That is, ||u|| is the square root of the sum of the squares of the components of u. Thus ||u|| ≥0, and ||u|| =0 if and only if u = 0.
Unit vector: A vector u is a called a unit vector if ||u|| =1 or, equivalently, if u.u = 1. For any non zero vector v in Rn, the vector

Is the unique unit vector in the same direction as v. The process of finding v ̅ from v is called normalizing in v.
Distance, angles, Projection:

Distance: The distance between vectors u=(a1, a2 , a3,…………….., an) and v= (b1, b2 , b3,……………..bn) in R n is denoted and defined by

Angle: The angle 0 between nonzero vectors u.v in Rn is defined by

This definition is well defined, since, by the Schwarz inequality

Note that if u.v = 0, then θ = 90^0.
Projections: The projection of a vector u onto a nonzero vector v is the vector denoted and defined by

Prove that, every basis of a vector space V has the same number of elements.

Problem: Prove that, every basis of a vector space V has the same number of ...