Equal Vectors:

Two vectors are equal if and only if their corresponding components are equal.
Component Form: v = < v1 ,v2 ,v3 >
magnitude: ||v|| = 

Examples:
(a)  Find ||v|| if v = < 3,1,-2 >.

      Answer:
     ||v|| = 

(b)  A (4,2,-6)  B(9,-1,3)  Write vector AB in component form and find its magnitude.

      Answer:
     Component Form: v = (9 - 4, -1 - 2, 3 - (-6)) = (5, -3, 9)
      Magnitude:||v|| = 
 

Unit Vectors:

A unit vector is a vector with length 1.  It can be found by dividing a vector by its magnitude.

Example:
Find the unit vector for v = < 2,-5,6 >

      Answer:
      Magnitude:||v|| = 
      Unit Vector:
 

Vector Arithmetic:

Let u = < -2,3,-1 >, v = < 6,2,3 >, and w = < 4,-1,5 >

Example:
Find each of the following:
(a)  u + v = < -2,3,-1 >  +  < 6,2,3 >  =  < -2 + 6, 3 + 2, -1 + 3 >  =  < 4, 5, 2 >
(b)  u - v = < -2,3,-1 >  -  < 6,2,3 >  =  < -2 - 6, 3 - 2, -1 - 3 >  =  < -8, 1, -4 >
(c)  2u = 2 < -2,3,-1 > = < -4, 6, -2 >
(d) -3v = -3 < 6,2,3 > = < -18, -6, -9 >
 

Dot Product:

Let u = < u1,u2,u3> and v = < v1,v2,v3>
The dot product of u and v is u× v = u1v1 + u2v2 + u3v3

Example:
Find the dot product of u and v if u = < 4,2,-3 > and v = < -3,5,1 >
      Answer:
      u× v = < 4,2,-3 > × < -3,5,1 > =   4(-3) + 2(5) + (-3)(1))   =   -12 + 10 - 3   =  -5
 
 

Properties of Vectors:

1.  u + v = v + u  (Commutative)
2.  (u + v) + w = u + (v + w)  (Associative)
3.  u + 0 = u  (Additive Identity)
4.  u + (-u) = 0  (Additive Inverse)
5.  c(du) = (cd)u  (c, d are scalars)
6.  (c + d)u = cu + du  (vector multiplication is distributive over scalars)
7.  c(u + v) = cu + cv  (scalar multiplication is distributive over vectors)
8.  1(u) = u
9.  0(u) = 0
10.  ||cu|| = |c| ||u||

Angle Between Two Vectors:
cos q =         u× v
                 ||u|| ||v||

Example:

Find the angle between u = < -2,7,4 > and v = < 4,1,-3 >

      Answer:
      cos q   =       < -2, 7, 4 > × < 4,1,-3 >       =              -2(4) + 7(1) + 4(-3)                 =            -8 + 7 - 12
                         ||< -2,7,4 >|| ||< 4,1,-3 >|| 
 

                 =          -13

 

      cos -1        -13        =  107.874o


Another form for this formula is  u× v = ||u|| ||v|| cos q
 

Orthogonal Vectors:

If two vectors u and v are orthogonal (perpendicular), then u× v = 0
The zero vector < 0,0,0 > is orthogonal to every vector because 0× u = 0
 

Standard Unit Vectors:

i = < 1,0,0 > , j = < 0,1,0 > and k = < 0,0,1 > are standard unit vectors.
The standard unit vectors are orthogonal to each other.
Standard unit vector notation: v = v1i + v2 j + v3k

Examples:
(a)  Name a vector parallel to < 4,3,1 >.

      Answer:
      Parallel vectors have the same slopes, so they are multiples of each other.  One example is < 8, 6, 2 >.
      Another is < 2, 3/2, 1/2 >.

(b)  Determine whether the following points lie on the same line. P< -2,7,4 >, Q< -4,8,1 >, R< 0,6,7 >

      Answer:
     The component form of  is < -6, 1, -3 > and the component form of vector  is < -12, -2, 6 >.
       -12 = (-2)6, -2 = (-2(1), and 6 = -2(3)
      Since one vector is a multiple of the other, the two vectors are parallel, and since they share a
      common point Q, they must be the same line.

(c)  The initial point of the vector v = 2i + 3j – 5k is P< 4,6,-3 >.  What is the terminal point of this vector?

      Answer:
      The question is:  What < a, b, c > allows < a, b, c > - < 4, 6, -3 > = < 2, 3, -5 >?
      < a - 4, b - 6, c - (-3) > = < 2, 3, -5 >
      a - 4 = 2   means a = 6
      b - 6 = 3   means b = 9
      c + 3 = -5 means c = -8
     so the terminal point of this vector is < 6, 9, -8 >


Problems(Questions are there, but no answers are posted)


Go to Precalculus Page
Go to Precalculus Lessons Page
Go to Analytic Geometry in Three Dimensions Lessons Page