Mathematical Investigations IV
Name:
Vectors
Getting To the Point
In Another Dimension
Drawing 3-D Vectors
In 3-D, we extend the unit vectors i and j and introduce k to give:
i = 1, 0, 0  , j =  0, 1, 0  , and k =  0, 0, 1 .
z
Sometimes, we show only the positive portion of
each axis. The x- and y-axes determine the
horizontal (ground) plane while the z-axis is vertical.
It may require some imagination, but the x-axis is
meant to be coming out of the plane towards the
reader. With negative coordinates, the negative
portion of the axes are shown as needed.
1
k
1
j
i
1
y
x
Examples:
(Notice that a “box” is dashed-in to help locate and visualize the vector.)
v = 2 i + 3 j + 4k = 2, 3, 4
w = 3 i – 5 j + k = 3, 5, 1
z
z


y
y
x
x
Now your turn: Draw the vectors indicated. (Be sure to include the dashed box)
1.
2.
v1 = 5i + 3j + 2k
v2 = –2i – 4j + 3k
z
z
y
y
x
x
Vectors 8.1
Rev. F08
Mathematical Investigations IV
Name:
Draw the vectors (and the axes). (Be sure to include the dashed box)
3.
4.
v3 = –3 i + 5 j + 4 k
v4 = 2 i + 5 j – 3 k
z
z
y
y
x
x
Fortunately, most of the formulas we have learned for vectors in two dimensions are easily
extended in a natural way to three dimensions. A couple of examples follow.
2-D
v  v1, v2 , w  w1, w2
v  v1, v2, v 3 , w  w1, w2, w3
3-D
Magnitude
2
v  v1  v 2
2
2
v  v1  v 2  v3
2
2
Dot product
v  w  v1w1  v2w2  v3 w3
v  w  v1w1  v2w2
Since the dot product and lengths are measured in the same way, then formulas for the cosine of
the angle between two vectors and the projection of one vector onto another are identical,
regardless of whether we are working in two or three dimensions.
Some problems:
5.
Let v =  3, 2, 4  and w =  –1, 3, –2  . Find:
v·v=
v·w=
|v|=
|w|=
v̂ (the unit vector in the same direction as v) =
ŵ (the unit vector in the same direction as w) =
Vectors 8.2
Rev. F08
Mathematical Investigations IV
Name:
z
6.
Let v = 2, 2, 1  and w =  5, 0, 0 .
Draw v and w in 3-space.
y
x
Find the angle between v and w to the nearest tenth of a degree.
7.
Let v =  2, 5, 9  and w =  5, 7, 10 . Find:
|v|
|w|
v·w
projwv
Find the angle between v and w to the nearest tenth of a degree.
8.
Find all values of a such that the vector q = < 2, a, –2> is perpendicular to the vector
p = < –3, a, 5 >
Vectors 8.3
Rev. F08
Mathematical Investigations IV
Name:
9.
Find scalars (a, b, c) such that v = ax + by + cz, where v = <3, –1, 4>,
x = <1, 2, 1>, y = <0, 4, –1>, and z = <–3, –2, 3>.
10.
Sketch the terms in the
3
sequence vt t 1 ,
z
where vt = < t, t2, 2t >.
y
x
Vectors 8.4
Rev. F08