Vector Algebra

53.4 Vector Algebra

Definitions

A vector is an entity which has a magnitude and direction. In two- and three-dimensional spaces, it is a directed line segment from one point to another. The projections of the vector on the x 1 ,x 2 , and x 3 axes

are a 1 ,a 2 , and a 3 and are called the vector components. It is represented by a column:

The length of the vector is designated by |a| and is given by

a § = 2 2 ( 12 a

a, b, g it makes with the axes or by their cosines. The latter are called direction cosines and are given by

A vector’s direction is given either by the angles

a a cos a a = ----- 1 cos b = ----- 2 cos g = ----- 3 (53.57)

It is evident that

2 cos 2 cos b + cos g = 1 (53.58) Generalizing a vector to n dimensions, we write

Vector Operations Equality.

a = b when a 1 = b 1 , a 2 = b 2 , L a , n = b n

Addition/Subtraction.

c = a ± b or

( a + b )c + = a + ( b + c )

Multiplication by a Scalar.

A scalar is a quantity which has magnitude but no direction, such as mass, temperature, time, etc., and will be designated by a lowercase Greek letter.

Any vector a is reduced to a unit vector a˚ when dividing its components by its length, which is a scalar, or a ∞ = aa § . The components of a˚ are the direction cosines of a. Unit vectors along the coordinate axes are called base or basis vectors and are given by

i = 0 j = 1 k = 0 (53.60)

Any vector in 3-space is uniquely expressed as

(53.61) The right-handed system introduced in Section 53.1 can be generalized for three vectors a, b, c. If they

are not coplanar, and they have the same initial point, then they are said to form a right-handed system if a right-threaded screw rotated through an angle less than 180˚ from a to b would advance in the direction c.

Vector Products Dot (or Scalar) Product.

ab ◊◊◊◊ =  a p b p = a 1 b 1 + a 2 b 2 + L + a n b n

This is also called the inner product. It is a scalar and has the following properties: This is also called the inner product. It is a scalar and has the following properties:

a ◊◊◊◊ ( b + c ) = ab ◊◊◊◊ + ac ◊◊◊◊

lab ( ◊◊◊◊ ) = ()b la ◊ = a ◊ ( lb ) = ( ab ◊◊◊◊ ) l

ii ◊◊◊◊ = jj ◊◊◊◊ = kk ◊◊◊◊ = 1 ij ◊◊◊◊ = jk ◊◊◊◊ = ki ◊◊◊◊ = 0

The dot product of a vector with itself is equal to the square of its length, or

1 + a 1 + La n = a (53.64) If q is the angle between two vectors a and b (in two- or three-dimensional space), it can be shown that

2 2 2 aa 2 ◊ = a

ab ◊◊◊◊ = ab cos q

It follows that if a is perpendicular to b, then ab ◊◊◊◊ = 0.

Cross (or Vector) Product. a ¥¥¥¥ b (read “a cross b”) is another vector c, which is perpendicular to both

a and b and in a direction such that a, b, c (in this order) form a right-handed system. The length of c is given by

(53.66) where q is the angle between a and b. This quantity is the area of the parallelogram determined by a and

c = a ¥¥¥¥ b = ab sin q

b . If a = a 1 i + a 2 j + a 3 k , and b = b 1 i + b 2 j + b 3 k , then c is given by the determinant

ijk

c = a ¥¥¥¥ b = a 1 a 2 a 3 (53.67)

It has the following properties

a ¥¥¥¥ b = – ( b ¥¥¥¥ a )

a ¥¥¥¥ ( b + c ) = a ¥¥¥¥ b + a ¥¥¥¥ c (observing the order)

a ◊◊◊◊ ( a ¥¥¥¥ c ) = 0

a ¥¥¥¥ b = a b – ( ab ◊◊◊◊ )

i ¥¥¥¥ i = j ¥¥¥¥ j = k ¥¥¥¥ k = 0

i ¥¥¥¥ j = kj ; ¥¥¥¥ k = ik ; ¥¥¥¥ i = j

For two nonzero vectors, if a ¥¥¥¥ b = 0 , then a and b are parallel.

Scalar Triple Product.

a ¥¥¥¥ bc ◊◊◊◊ = b 1 b 2 b 3 (53.69)

is a scalar which is equal to the volume of the parallelepiped determined by a, b, c. If it is zero, then the three vectors are coplanar. It has the following properties:

a ¥¥¥¥ bc ◊◊◊◊ = b ¥¥¥¥ ca ◊◊◊◊ = c ¥¥¥¥ ab ◊◊◊◊

a ¥¥¥¥ bc ◊◊◊◊ = ab ◊◊◊◊ ¥¥¥¥ c

Planes and Lines

If p 0 is a given point in a plane, n is a nonzero vector normal to the plane, and p is any point in the plane, then the equation of the plane takes the form

pn ◊◊◊◊ – p 0 ◊◊◊◊ n = 0 (53.70) Let n = Ai + Bj + Ck, p 0 =X 0 i +Y 0 j +Z 0 k , and p = Xi + Yj + Zk. Then Eq. (53.70) becomes AX ( – X 0 )BYY + ( – 0 )CZZ + ( – 0 ) = 0 (53.71)

( p – p 0 )n ◊◊◊◊ = 0 or

or

AX + BY + CZ + D = 0

where D = – (AX 0 + BY 0 + CZ 0 ). Two planes are parallel when they have a common normal vector n, and are

perpendicular when their normals are, or n 1 ·n 2 = 0.

If p 0 represents a given point on a line, p any other point on the line, and v is a given nonzero vector parallel to the line, then

(53.72) is an equation of the line. In component form, it yields three scalar equations describing the parametric

p = p 0 + lv

form ( l is the running parameter):

If l is eliminated, one gets the usual two-equation form of a straight line in space; see Eq. (53.54).