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VG.2 Vectors

Vector- Term used by scientists to indicate a quantity that has both magnitude and direction. The length of the arrow indicates the magnitude and the arrow points in the direction of the vector.

Representation of Vectors

Vectors are Represented as follows:

The numbers $a_{1}$ , $a_{2}$ , and $a_{3}$ are called components of $\QTR{bf}{a}$ .

A vector is a directed line segment from any point to . When is seen alone, it is understood to be a vector that starts at the origin and ends at the point .

Using Two Points to find a vector that begins at the origin

Given the points $A(x_{1},y_{2})$ and $B(x_{2},y_{2})$ the vector a with a representation of is

Given the points and the vector a with a representation of is

Finding the magnitude (or length) of a vector

The length of the two-dimensional vector is

The length of the three-dimensional vector is

The only vector with the length 0 is the zero vector (also ). This is the only vector with no specific direction.

Adding vectors

If and then the vector $\QTR{bf}{a+b}$ is defined by

Similarly, for three-dimensional vectors,

Multiplying a Vector by a scalar

If is a scalar and then the vector $c\QTR{bf}{a}$ is defined by

Also for three-dimensional vectors,

Subtracting vectors

If and then the vector $\QTR{bf}{a+(-b)}$ is defined by

Similarly, for three-dimensional vectors,

Properties of Vectors

MATH

Three special vectors

In three-dimensions:

Also in two-dimensions,

The three vectors $\QTR{bf}{i,j,k}$ can be used to define the vector or the vector

Also in two-dimensions,

The unit vector

For any nonzero vector a we have:

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