Module 8 Lesson 3: Vectors in 3D
In Lesson 1-2, we worked strictly with two-dimensional vectors (vectors in the xy plane).
Here in Lesson 3, we will extend our knowledge to three-dimensional vectors (vectors in
the xyz plane). Most of the “operations” and properties of 2D vectors are very similar to
those of 3D vectors.
The only difference is that we will be introduced to another method of multiplying two 3D
vectors, called the cross product. To make this simple, we will learn
how to find the cross product using matrices on our calculator.
Let’s get started.
Vectors in 3D (Vectors in Space)
All of our work with vectors thus far has been in the two
dimensional plane. In three-dimensional space (the x-y-z plane) a
vector is denoted by an ordered triple (rather than an ordered pair
as in 2D):
𝐯 = ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩
Note the vector v shown in red in the figure to the right.
If v is the directed line segment with initial point 𝑃(𝑝1 , 𝑝2 , 𝑝3 )
and terminal point 𝑄(𝑞1 , 𝑞2 , 𝑞3 ), then the component form is as
follows:
𝐯 = ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩ = ⟨𝑞1 − 𝑝1 , 𝑞2 − 𝑝2 , 𝑞3 − 𝑝3 ⟩
The linear combination notation for v would then be
𝐯 = 𝑣1 𝐢 + 𝐯𝟐 𝐣 + 𝑣3 𝐤
The magnitude of v can be found using the following formula:
‖𝐯‖ = √𝑣1 2 + 𝑣2 2 + 𝑣3 2
Ex. Find the component form and the magnitude of vector w with initial point (3, 4, 2)
and terminal point (3, 6, 4).
The component form is:
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𝐰 = ⟨3 − 3, 6 − 4, 4 − 2⟩ = ⟨0, 2, 2⟩
The magnitude is:
‖𝐰‖ = √02 + 22 + 22 = √8 = 2√2
Here are a few more properties that hold true for 3D vectors.
1. Two vectors are equal if and only if their corresponding components are equal.
𝐯
2. A unit vector u in the direction of v is 𝐮 = ‖𝐯‖ , 𝐯 ≠ 0.
3. The sum of 𝐮 = ⟨𝑢1 , 𝑢2 , 𝑢3 ⟩ and 𝐯 = ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩ is
𝐮 + 𝐯 = ⟨𝑢1 + 𝑣1 , 𝑢2 + 𝑣2 , 𝑢3 + 𝑣3 ⟩
4. The scalar multiple of the real number c and 𝐮 = ⟨𝑢1 , 𝑢2 , 𝑢3 ⟩ is
𝑐𝐮 = 𝑐⟨𝑢1 , 𝑢2 , 𝑢3 ⟩ = ⟨𝑐𝑢1 , 𝑐𝑢2 , 𝑐𝑢3 ⟩
5. The dot product of 𝐮 = ⟨𝑢1 , 𝑢2 , 𝑢3 ⟩ and 𝐯 = ⟨𝑣1 , 𝑣2 , 𝑣3 ⟩ is
𝐮 ∙ 𝐯 = 𝑢1 ∙ 𝑣1 + 𝑢2 ∙ 𝑣2 + 𝑢3 ∙ 𝑣3
Ex. Given 𝐮 = ⟨0, 3, −2⟩ and 𝐯 = ⟨4, 2, 3⟩, find the following.
a. Find 2𝐮 + 𝐯.
2𝐮 + 𝐯 = 2⟨0, 3, −2⟩ + ⟨4, 2, 3⟩
= ⟨0, 6, −4⟩ + ⟨4, 2, 3⟩
= ⟨0 + 4, 6 + 2, −4 + 3⟩
= ⟨4, 8, −1⟩
b. Find the dot product of 𝐮 ∙ 𝐯.
⟨0, 3, −2⟩ ∙ ⟨4, 2, −3⟩ = 0 ∙ (4) + 3(2) + (−2)(−3)
= 0+6+6
= 12
c. Find the unit vector for v.
⟨4, 2, 3⟩
𝐯
=
‖𝐯‖ √42 + 22 + 32
=
=⟨
2
4
⟨4, 2, 3⟩
√29
2
3
⟩
√29 √29 √29
,
,
=⟨
4√29 2√29 3√29
⟩
,
,
29
29
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The Cross Product
Another method of multiplying two vectors is by the cross product. Many applications in
physics, engineering, and geometry involve finding a vector in space that is orthogonal (or
perpendicular) to two given vectors. In order to find such a vector, one must use the
cross product, and it is most conveniently defined and calculated using matrices with your
TI Graphing Calculator.
Let’s first look at the set up of how to find the cross product. Notice that we use the “x”
or “cross” multiplication symbol here. When reading 𝐮 × 𝐯 aloud, we say “u cross v”.
Given two vectors u and v, we can set up a matrix as follows:
𝐢
𝐮 × 𝐯 = |𝑢1
𝑣1
𝐣
𝑢2
𝑣2
𝐤
𝑢3 |
𝑣3
Put u in row 2.
Put v in row 3.
Then finding the determinants and arranging the values as follows:
𝑢2
= |𝑣
2
𝑢3
𝑢1
𝑣3 | 𝐢 − |𝑣1
𝑢3
𝑢1
𝑣3 | 𝐣 + |𝑣1
𝑢2
𝑣2 | 𝐤
Let’s see how to find the cross product using our TI Graphing Calculator.
Ex. Find 𝐮 × 𝐯 given 𝐮 = 𝐢 + 2𝐣 + 𝐤 and 𝐯 = 3𝐢 + 𝐣 + 2𝐤.
Step 1: Write your 3 × 3 matrix…
𝐢
𝐮 × 𝐯 = |1
3
𝐣 𝐤
2 1|
1 2
Step 2: Write the cross product in the form of your three determinants…
1 1
1 2
2 1
𝐮 × 𝐯=|
|𝐢 − |
|𝐣 + |
|𝐤
3 2
3 1
1 2
3
Notice the minus sign
in front of the j
component.
Step 3: Enter each of these three matrices into your calculator. For convenience, choose
to put them in as Matrices [A], [B], and [C].
Scroll to the right and choose EDIT. We will be changing matrices [A], [B], and [C] to 2 x 2
matrices.
Select Matrix [A], change its dimensions to 2 x 2 and enter in the matrix for
component i. Press 2nd  MARTRIX.
Select Matrix [B], change its dimensions to 2 x 2 and enter in the matrix for
component j.Press 2nd  MARTRIX.
Select Matrix [C], change its dimensions to 2 x 2 and enter in the matrix for
component k.
Once you have all of your matrices entered, we can find the determinant for each, and
thus find our components for I, j, and k.
Press 2nd  QUIT in order to return to your Home screen.
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Step 4: In your Home screen, press 2nd  MATRIX again. Scroll right and select MATH.
Select 1:det(.
Step 5: Next, press 2nd  MATRIX. Don’t scroll to the right. Instead, simply choose
option 1: [A] 2x2 and press ENTER.
Step 6: Repeat Steps 4-5 to find the determinants for matrices [B] and [C].
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Step 7: Rewrite these values as the result of the cross product 𝐮 × 𝐯.
𝐮 × 𝐯 = 3𝐢 − (−1)𝐣 + (−5)𝐤
= 3𝐢 + 𝐣 − 5𝐤
Done!
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