Vectors

Vector is a quantity specified by magnitude plus direction in space

Properties of vectors

Any vector is uniquely specified by its three components x, y, z which are projections of the vector on coordinate axes with unit vectors (such that )

The notation of vector with coordinates has the form

is

Magnitude of vector (or its length) is defined by Pythagorean Theorem

|| =

Adding vectors

Adding of vectors and is sum vector

The sum vector is determined by Parallelogram Rule of addition

Magnitude of sum vector is defined by Law of Cosines

where:

and are magnitudes of the vectors and

is angle between them

Components of sum vector:

x = x₁+ x₂

y = y₁+ y₂

z = z₁+ z₂

The parallelogram rule of addition is partial case of general Polygon Rule used for adding several vectors

Subtracting vectors

Subtracting of vectors and is vector difference

The vector difference is determined by Triangle Method of subtraction

Magnitude of vector difference

Components of vector difference:

x = x₁ - x₂

y = y₁ - y₂

z = z₁ - z₂

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