Vector Addition and Scalar Multiplication
Tutorial on the addition and scalar multiplication of vectors.
Vectors are mathematical quantities used to represent concepts such as force or velocity which have both a magnitude and a direction.
Components of a VectorThe component form of vector v with initial point A(a1,a2) and terminal point B(b1,b2) is given by
If a vector is given by v = < v1 , v2 > , it magnitude || v || is given by
Example 1: Find the components and the magnitude of vector v with initial point A(2,3) and terminal point B(4,5).
Solution to example 1:
Use above definition to find vector v
v = < v1 , v2 > = < b1 - a1 , b2 - a2 >
= < 4 - 2 , 5 - 3 > = < 2 , 2 >
and its magnitude || v ||
|| v || = √(v1 2 + v2 2)
= √(2 2 + 2 2) = √(8) = 2 √(2)
Scalar Multiplication of a VectorThe scalar multiplication of vector v = < v1 , v2 > by a real number k is the vector k v given by
k v = < k v1 , k v2 >
Addition of two Vectors
The addition of two vectors v(v1 , v2) and u (u1 , u2) gives vector
v + u = < v1 + u1 , v2 + u2>
Below is an html5 applets that may be used to understand the geometrical explanation of the addition of two vectors. Enter components of vectors A and B and use buttons to draw, add, zoom in and out as well as translate the system of axes.
An online vector addition calculator may be used to check any answers to examples below.
Example 2: Vectors v and u are given by their components as follows
Example 3: v and u are vectors given by
Answers to above exercises
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Updated: 27 November 2007 (A Dendane)