1 Chalkboard
Figure 1: Determinant in space
Figure 2: Volume of parallelepiped
2 What is the volume between 3 vectors?
2.1 Front
What is the volume between 3 vectors?
2.2 Back
- With determinants
- \(\abs{\det(\vb{A}, \vb{B}, \vb{C})}\)
- With cross product
- Volume = area(base) height
- \(V = \abs{\vb{A} \cross \vb{B}} (\vb{C} \cdot \hat{n})\)
- \({\displaystyle \hat{n} = \frac{\vb{A} \cross \vb{B}}{\abs{\vb{A} \cross \vb{B}}}}\)
- \({\displaystyle V = \abs{\vb{A} \cross \vb{B}} (\vb{C} \cdot \frac{\vb{A} \cross \vb{B}}{\abs{\vb{A} \cross \vb{B}}}) = \vb{C} \cdot (\vb{A} \cross \vb{B})}\)
3 What happens if \(\vb{a} \cross \vb{b} = 0\)?
3.1 Front
What happens if $\vb{a} \cross \vb{b} = 0$?
3.2 Back
We can say that this 2 vector are parallel
4 How we can compute cross product and why?
4.1 Front
How we can compute cross product and why?
4.2 Back
- Compare geometrically the volume of 3 vectors
- \(\det(\vb{A}, \vb{B}, \vb{C}) = \vb{A} \cdot (\vb{B} \cross \vb{C})\)
- \({\displaystyle \vb{A} \cross \vb{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}}\)
- \(\abs{\vb{A} \cross \vb{B} = \abs{\vb{A}} \abs{\vb{B}} \sin \theta}\)
The result is a vector
5 Which is the direction of the vector from cross product?
5.1 Front
Which is the direction of the vector from cross product?
5.2 Back
- Result is orthogonal to the plane of the 2 vectors
- Need to use right hand method
- Extend the right hand along 1st vectors
- Move your finger to the other vector
- Your thumb will point to the direction of resulting vector
6 What can we say if \(\abs{\det(\vb{A}, \vb{B}, \vb{C})}=0\)?
6.1 Front
What can we say if $\abs{\det(\vb{A}, \vb{B}, \vb{C})}=0$?
6.2 Back
The volume of this parallelepiped with these vectors as edges is \(0\). This means all three origin vectors lie in a plane.
7 What does means \(\abs{\vb{v}}^{2}\)
7.1 Front
What does means $\abs{\vb{v}}^{2}$
Where \(\vb{v}\) is a vector
7.2 Back
\(\abs{\vb{v}}^{2} = \vb{v} \cdot \vb{v}\)
8 Determinant - ij-cofactor
8.1 Front
Determinant - ij-cofactor
What is a ij-cofactor
8.2 Back
\(A_{ij} = (-1)^{i+j}|A_{ij}|\)
9 Determinant - ij-entry
9.1 Front
Determinant - ij-entry
What is the ij-entry?
9.2 Back
\(a_{ij}\) is the number in the i-th row and j-th column
10 Determinant - ij-minor
10.1 Front
Determinant - ij-minor
What is the ij-minor?
10.2 Back
\(|A_{ij}|\) is the determinant that’s left after deleting from \(|A|\) the row and column containing \(a_{ij}\)
11 What is the method of determinant computation?
11.1 Front
What is the method of determinant computation?
11.2 Back
- It’s called the Laplace expansion by cofactors
12 How compute a determinant through the Laplace expansion?
12.1 Front
How compute a determinant through the Laplace expansion?
Explain how compute a determinant using laplace expansion by cofactors
12.2 Back
- Choose a row or a column
- Multiply each entry \(a_{ij}\) in that row (or column) by its cofactor \(A_{ij}\)
- Add all resulting numbers
Examples
- 1st rows of 3x3 determinant
- \(a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}\)
- j-th column of 3x3 determinant
- \(a_{1j}A_{1j} + a_{2j}A_{2j}+a_{3j}A_{3j}\)
13 What does mean \(A_{12}\)?
13.1 Front
What does mean $A_{12}$?
13.2 Back
This is the 12-cofactor of a determinant, which value is \((-1)^{1+2}|A_{12}|\)
\(|A_{12}|\) is the ij-minor is a determinant after removing 1st row and 2nd column
14 What does mean \(\abs{A_{23}}\)
14.1 Front
What does mean $\abs{A_{23}}$
14.2 Back
This is determinant resulting from removing the 2nd row and 3rd column of the matrix \(A\)
15 What does mean \(\abs{A}=0\)?
15.1 Front
What does mean $\abs{A}=0$?
15.2 Back
- One row or column is all zero
- If two rows or two columns are the same
If \(A\) is a matrix that represent 3 vectors \(3 \times 3\), it means that these 3 vectors are coplanar. This means also that the volume of this parallepiped is 0
16 What does mean that \(\abs{A}\) is multiplied by \(c\)?
16.1 Front
What does mean that $\abs{A}$ is multiplied by $c$?
16.2 Back
- Every element of some row or column is multiply by \(c\)
17 How we can change the sign of a determinant?
17.1 Front
How we can change the sign of a determinant?
17.2 Back
- We interchange two rows or two columns
18 When 2 determinants does have the same value?
18.1 Front
When 2 determinants does have the same value?
18.2 Back
- If we add to one row (or column) a constant multiple of another row (or column)
