White Plains High School
Mr.Stanton
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SWBAT to find the cross product of two vectors
𝑎
Recall: The determinant of a 2x2 matrix is defined as: det A = |𝐴| = |
𝑐
Do Now:
2
1. Find the Determinant of A = [
3
5
]
−1
𝑏
| = 𝑎𝑑 − 𝑏𝑐.
𝑑
1 2 3
2. Find the Determinant of B =[4 0 2]
3 2 1
Here is a different way to find the determinant of a 3 x 3 that will save some time…
If
𝑎
A=[𝑑
𝑔
A a
𝑏
𝑒
ℎ
𝑐
𝑓] then you find |𝐴| by the following algorithm:
𝑖
d f
d e
e f
b
c
g i
g h
h i
Notes:
1)The little determinants are called minor determinants.
2) Minor determinants are found using the “2 pencil test”.
3) The middle term is subtracted.
Use this method to determinants of the following matrices:
−3 4 −1
−1 −1 0
a) [−2 −2 2 ]
b) [−3 −3 4]
−5 4
1
5
5 4
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−2 2 −5
c) [ 2 2 5 ]
1 4 4
White Plains High School
Mr.Stanton
Document1
The CROSS PRODUCT
Recall the scalar product:
Recall the dot product:
c  v   c a, b  ca, cb
u  a , b , v  c, d ,
u  v  ac  bd  a b cos 
The 3rd way multiplication can be used with vectors is called the CROSS PRODUCT.
Here is the formula, but it is NOT practical to memorize it.
a  ax, ay, az , b  bx, by, bz , a  b  aybz  azby , azbx  axbz , axby  aybx
The Cross Product of two vectors is a vector that is orthogonal (perpendicular, but not co-planar) to
both vectors. The direction of the cross product vector follows the Right Hand Rule…
1) Have the fingers of your right hand point along 1st vector, a
2) Curl fingers through smallest angle theta in the direction of 2 nd vector, b
3) Your right thumb will point in the direction of the cross product vector.
Reflect: Is cross product commutative: Does
ab  b a ?
To find the cross product without having to memorize the formula in the box above, we will use
matrices and determinants.
Example:
a  5,1, 4 , b  1, 0, 2 , find
ab
̂ represent the first row of a 3x3 matrix with the two vectors a
Step 1: Let the unit vectors 𝑖̂ , 𝑗̂ , 𝑘
and b representing the remaining two rows.
i j k
5 1 4
1 0 2
Step 2: The determinant of the 3x3 matrix will be the cross product!
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White Plains High School
Mr.Stanton
Document1
Find the cross products:
u  2,3, 4 , v  1, 6, 4
The magnitude of the cross product will equal the area of the parallelogram formed by the two
vectors!
Find u  v
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White Plains High School
Mr.Stanton
Document1
HW 7.8 {1-15 only}
Note: In 5-10, i, j, k represent the unit vectors i  1, 0, 0 ; j  0,1, 0 ; k  0, 0,1
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