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.
The figure below shows vector v with initial point A and terminal point B.

initial and terminal points of vector

Components of a Vector

The component form of vector v with initial point A(a1,a2) and terminal point B(b1,b2) is given by
v = < b1 - a1 , b2 - a2 >

If a vector is given by v = < v1 , v2 > , it magnitude || v || is given by
|| v || = √(v1 2 + v2 2)

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 Vector

The 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.
Components of vector A
Components of vector B
Zoom
Translate origin
>

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
v = < -2 , 3> and u = < 4 , 6>
Find each of the following vectors.
1 : v + 2 u
2 : u - 4 v
Solution to example 2:
First carry out the scalar multiplication 2 u then the addition

1 : v + 2 u = <-2 , 3> + 2 <4 , 6> = <-2 , 3> + <8 , 12>
= <6 , 15>
2 : u - 4 v = <4 , 6> + (- 4) <-2 , 3> = <4 , 6> + <8 , -12>
= <12 , -6>

Example 3: v and u are vectors given by
v = < 1 , -2> and u = < u1 , u2>
Find components u1 and u2 of vector u so that v + 3 u = 0.
Solution to example 3:
We first obtain v + 3 u in terms of u1 and u2

v + 3 u = <1 , -2> + 3 <u1 , u2>

= <1 , -2> + <3 u1 , 3 u2>
= <1 + 3 u1 , -2 + 3 u2>
For the above vector to be equal to vector 0, its two components have to be equal to 0, hence
1 + 3 u1 = 0 and -2 + 3 u2 = 0
Solve the first equation for u1 and the second equation for u2
u1 = -1 / 3 and u2 = 2 / 3

Exercises
1. Given vectors
v = <-3 , 2> and u = <-2 , 0>,
find the following vectors.
- v + 2 u , v - (1/2) u
2. Vectors v and u are given by
v = <4 , 1> and u = <u1 , u2>,
find components u1 and u2 so that 2 v - 3 u = 0 .

Answers to above exercises
1.
- v + 2 u = <- 1 , -2>,
v - (1/2) = <- 2 , 2>,
2.
u1 = 8 / 3
u2 = 2 / 3

More pages and references related to vectors.



Vector Addition - HTML5 Applet

Vector Subtraction - HTML5 Applet

Vector Calculators.
More on Vectors. and vectors in Physics


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Updated: 27 November 2007 (A Dendane)
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