Calculus 3-The Cross Product of Vectors in-R3, guaranteed 100% Pass

1


The Cross Product of Vectors in ℝ3

The cross product is only defined for vectors in ℝ3 and unlike the dot product, the
answer is a vector in ℝ3 .


Def. If 𝐴⃑ = < 𝑎1 , 𝑎2 , 𝑎3 > , 𝐵
⃑⃑ = < 𝑏1 , 𝑏2 , 𝑏3 >, then:

⃑⃑ × 𝐁
𝐀 ⃑⃑⃑ = < 𝑎2 𝑏3 − 𝑎3 𝑏2 , 𝑎3 𝑏1 − 𝑎1 𝑏3 , 𝑎1 𝑏2 − 𝑎2 𝑏1 >


= (𝑎2 𝑏3 − 𝑎3 𝑏2 )𝑖⃑ + (𝑎3 𝑏1 − 𝑎1 𝑏3 )𝑗⃑ + (𝑎1 𝑏2 − 𝑎2 𝑏1 )𝑘⃑⃑.



Recall:
𝑎 𝑏 𝑎 𝑏
𝑑𝑒𝑡 ( )=| | = 𝑎𝑑 − 𝑏𝑐
𝑐 𝑑 𝑐 𝑑


𝑎1 𝑎2 𝑎3
𝑏 𝑏3 𝑏 𝑏3 𝑏 𝑏2
|𝑏1 𝑏2 𝑏3 | = 𝑎1 | 2 | − 𝑎2 | 1 | + 𝑎3 | 1 |.
𝑐2 𝑐3 𝑐1 𝑐3 𝑐1 𝑐2
𝑐1 𝑐2 𝑐3



1 −2 3
1 0 2 0 2 1
Ex. |2 1 0| = 1 | | − (−2) | | + 3| |
2 1 0 1 0 2
0 2 1

= 1(1 − 0) + 2(2 − 0) + 3(4 − 0)

= 1 + 4 + 12 = 17.

, 2


We can rewrite 𝐴⃑ × 𝐵
⃑⃑ as:

(𝑎2 𝑏3 − 𝑎3 𝑏2 )𝑖⃑ + (𝑎3 𝑏1 − 𝑎1 𝑏3 )𝑗⃑ + (𝑎1 𝑏2 − 𝑎2 𝑏1 )𝑘⃑⃑

𝑎2 𝑎3 𝑎1 𝑎3 𝑎1 𝑎2
= |𝑏 | 𝑖
⃑ − | | 𝑗
⃑ + | ⃑⃑
2 𝑏3 𝑏1 𝑏3 𝑏1 𝑏2 | 𝑘

𝒊⃑ 𝒋⃑ ⃑⃑
𝒌
⃑𝑨
⃑⃑ × ⃑𝑩
⃑⃑ = |𝒂𝟏 𝒂𝟐 𝒂𝟑 |.
𝒃𝟏 𝒃𝟐 𝒃𝟑

Ex. If 𝐴⃑ = < 2, 1, −2 > , 𝐵
⃑⃑ = < −3, 2, 1 > find 𝐴⃑ × 𝐵
⃑⃑.



𝑖⃑ 𝑗⃑ 𝑘⃑⃑
𝐴⃑ × 𝐵
⃑⃑ = | 2 1 −2|
−3 2 1

1 −2 2 −2 ⃑⃑ | 2 1
= 𝑖⃑ | | − 𝑗⃑ | |+𝑘 |
2 1 −3 1 −3 2

⃑⃑ (4 − (−3))
= 𝑖⃑(1 − (−4)) − 𝑗⃑(2 − 6) + 𝑘

⃑⃑
= 5𝑖⃑ + 4𝑗⃑ + 7𝑘




Ex. Show 𝐴⃑ × 𝐴⃑ = 0
𝑖⃑ 𝑗⃑ 𝑘⃑⃑ 𝑎2 𝑎3 𝑎1 𝑎3 𝑎 𝑎2
⃑ ⃑
𝐴 × 𝐴 = |𝑎1 𝑎3 | = 𝑖⃑ |𝑎 | − 𝑗
⃑ | | + ⃑⃑ | 1
𝑘
𝑎2
2 𝑎3 𝑎1 𝑎3 𝑎1 𝑎2 |
𝑎1 𝑎2 𝑎3


= (0)𝑖⃑ − (0)𝑗⃑ + (0)𝑘⃑⃑ = 0
⃑⃑.