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19 Cards in this Set
- Front
- Back
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Determinant of a 2x2 matrix |
|{a₁₁,a₁₂},{a₂₁,a₂₂}| determinant= (a₁₁×a₂₂)-(a₁₂×a₂₁) |
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Process of finding determinant of an nxn matrix |
Chose one row or one column, then take the determinant without the row/column of the value you had originally chosen, repeat until 2×2 matrices are obtained |
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Effect on determinant if you interchange rows or columns in a matrix |
The determinant is multiplied by -1 |
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Effect on determinant if there is a scalar in a row or column |
We can factor the scalar out of the row or column |
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Effect on determinant if any row or column consists completely of 0 |
The determinant is 0 |
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Effect on determinant if we change a row (or column) by adding another row (or, respectively, column) |
The value of the determinant remains the same |
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Definition: The cross product (a×b) |
Suppose that a= a₁i + a₂j +a₃k, and b=b₁i + b₂j +b₃k are vectors in R³. The cross product or vector product of a and b (a×b) is defined to be the vector a×b =|{i,j,k},{a₁,a₂,a₃},{b₁,b₂,b₃}| |
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Properties of the cross product (4) |
1. a×b = -(b×a) 2. a×(βb +γc) = β(a×b) + γ(a×c) 3. (αa + βb)×c = α(a×c) + β(b×c) 4 a×b =0 if and only if a and b are parallel or a or b is zero |
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What happens when you take cross product of the same vector |
a×a= -a×a because of the property so a×a=0 |
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Cross products of unit vectors (6) |
i×i=0 j×j=0 k×k=0 i×j=k j×k=i k×i=j |
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How is the vector a×b related directionally to any vector in the plane spanned by a and b |
It is orthogonal to any vector in the plane spanned by a and b Also, it is perpendicular to a and b |
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Length of a×b= ‖a×b‖= |
=‖a‖ ‖b‖ |sinθ| |
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The area of the parallelogram spanned by a and b |
‖a‖ ‖b‖ sinθ |
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Geometry of 2×2 Determinants |
The absolute value of the determinant |{a₁,a₂},{b₁,b₂}| is the area of the parallelogram whose adjacent sides are the vectors a=a₁i + a₂j and b=b₁i + b₂j. The sign of the determinant is positive when the angle from a to b is less than π, rotating in the counterclockwise direction |
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Geometry of 3×3 Determinants |
The absolute value of the determinant D= |{a₁,a₂,a₃}, {b₁,b₂,b₃}, {c₁,c₂,c₃}| is the volume of the parallelepiped whose adjacent sides are the vectors that correlate in the matrix |
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Equation of a Plane in Space |
The equation of the plane P through (x₀,y₀,z₀) that has a normal vector n=Ai+Bj+Ck is A(x-x₀) +B(y-y₀) +C(z-z₀)=0 That is (x,y,z) ∈P if and only if Ax+By+Cz+D=0 where D= -Ax₀-By₀-Cz₀ |
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Determining A,B,C,D for the plane P |
x,y,z can satisfy the Ax+By+Cz+D=0 equation if and only if it satisfies (λA)x+(λB)y+(λC)z+(λD)=0 for any constant λ≠0. Also, if both A,B,C,D and A',B',C',D' determine P, then A=λA', B=λB' C=λC', D=λD' for a scalar λ. |
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Qualifications for two planes being called parallel |
When their normal vectors are parallel |
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Distance from a Point (x₁,y₁,z₁) to a Plane (Ax+By+Cz+D=0) |
Distance = |Ax₁+By₁+Cz₁+D|/√(A²+B²+C²) |
