Vector Mathematics

Question 1

Dot Product

Answer 1

u · v = || u || || v || cos a

or

u · v = u₁v₁ + u₂v₂

• u and v are non-zero vectors
• 0 <= a <= π
• u · 0 = 0

Question 2

Scalar Projection of v onto u

Answer 2

comp_uv = || v || cos a = ( u·v )/|| u || = û·v

• u and v are non-zero vectors
• 0 <= a <= π
• This is a number that represents the magnitude of the Vector Projection of v onto u.
• “comp” is short for “component.” This is the component of u in the direction of v.

Question 3

Vector Projection of v onto u

Answer 3

proj_uv = (comp_uv)û

• The vector in the direction of u with a magnitude of comp_uv, provided the angle between them is acute. Otherwise it points in the direction of -u.

Question 4

Properties of the Dot Product

Answer 4

Suppose u, v, and w are vectors and a and b are scalars. Then

• u · u = || u ||^2
• u · v = u · v
• (au) · (bv) = ab(u · v)
• u · (v + w) = u · v + u · w = (v + w) · u

Question 5

Angle Between Vectors

Answer 5

The angle between the non zero vectors u and v is

a = cos^-1[u · v / (|| u || || v ||)]

Question 6

The Cross Product

Answer 6

u x v = || u || || v || (sin a) n

• u and v are non-zero vectors
• 0 <= a <= π
• n is a unit vector that is orthogonal to u and v and points in the direction determined by the right-hand rule.

Question 7

Properties of the Cross Product

Answer 7

Suppose u, v, and w are 3-vectors, a and b are scalars, and a is the angle between u and v. Then…