Mathematics 1

Scalar Dot Product in terms of vectors’ components

• At first let us find scalar product of unit vectors
• Since the standard unit vectors are orthogonal the dot product between a pair of distinct standard unit vectors is zero:

ij= jk=ik=0

• The dot product between a unit vector and itself:

ii= jj=kk=1

• Then using the dot product listed in the previous slide we can write:
• a.b = (axi + ayj + azk)(bxi + byj + bzk)= =axbxii+aybxji+azbxki +axbyij+ aybyjj+azbykj + +azbxki+azbykj+azbzkk=axbx +ayby+azbz

Scalar Dot Product -examples

Example 1

• If a=(6,−1,3), for what value of c is the vector b=(4,c,−2) perpendicular to  a?
• Solution:
• For  a  and  b to be perpendicular, we need their dot product to be zero. Since:
• a.b=6*4+c*(-1) +3*(-2) = -c+18=0;
• c=18
• Example 2
• Determine the angle between ν =(2, 5, 3) and ω =(1, - 2, 4)
• Solution: = ===0.1416
• =1.5359rad

•  

Vector (Cross) Product

• Suppose that a and b are non-zero 3D vectors (the vector product is only defined for 3D vectors).
• They determine a plane in which each lies. The vector product (or cross product) is defined by
• a b = sin()n;
• where 0 is the
• angle between the positive
• directions of a and b, and
• n is the unit vector
• perpendicular to the plane
• of a and b, with direction
• determined by the
• right-hand-rule.
• (right handed screw)

•  

Vector Product properties

• We can summarise the vector product properties as follows:(here a , b and c are vectors and is a scalar):
• a b = sin()n
• | a × b| = |a||b| sin θ
• The vector a b is perpendicular to both a and b.
• a a=0
• If a × b = 0, then a and b are parallel.
• And conversely :
• Two nonzero vectors a and b are parallel if and only if