Working with Vectors

If we divide an arbitrary vector a by its magnitude we obtain a (dimensionless) vector in the same direction as a and of magnitude 1 (a dimensionless number). Such a vector is known as a unit vector in the direction of vector a and is denoted by â = a/|a|

A zero vector is defined so that a + (-1)a = 0

Addition of vectors is both commutative and associative i.e. a + b = b + a and a + (b + c) = (a + b) + c

For any scalar k, ka + kb = k(a + b)

A vector may be resolved into component vectors along appropriately chosen directions. Given a vector a, its orthogonal component vectors parallel and normal to a direction inclined at an angle θ to a are of magnitude,

|ap| = |a| cos θ and |an| = |a| sin θ

The Cartesian unit vectors in the directions of the Cartesian axes x, y, z are denoted by i, j, k.

A vector a can be expressed in Cartesian form as a = axi + ayj + azk

The scalars ax, ay, az are called the Cartesian scalar components of a, whereas the vectors axi, ayj, azk are called the Cartesian component vectors of a.

The magnitude of the vector a is given by |a| = (ax2 + ay2 + az2)1/2

The operation of scaling and vector addition take the following Cartesian algebraic forms (for any scalar α and vectors a and b);

αa = αaxi + αayj + αazk

and

a + b = (ax + bx)i + (ay + by)j + (az + bz)k

A vector represented by a = axi + ayj + azk can also be represented by the abbreviated notation of a tripled pair a = (ax, ayaz).

The operation of scaling and vector addition take the following abbreviated forms (for any scalar α and vectors a and b),

αa = (αax, αay, αaz)

and

a + b = (ax + bx, ay + by, az + bz)

Vectors may be used to determine the position of points relative to a chosen origin , and they are then known as position vectors.

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~ by jamesdow2013 on April 14, 2013.

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