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15 Cards in this Set
- Front
- Back
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Standard basis vectors |
i: vector with components (1,0,0) j: vector with components (0,1,0) k: vector with components (0,0,1) If a vector has components (a₁,a₂,a₃) it can be written as v= a₁i + a₂j + a₃k |
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Parametric Equation of a Line: Point-Point Form |
The parametric equations of the line L through the points P= (x₁,y₁,z₁) and Q=(x₂,y₂,z₂) are x= x₁ + (x₂-x₁)t y= y₁ + (y₂-y₁)t z= z₁ + (z₂-z₁)t where (x,y,z) is the general point of L, and the parameter t takes on all real values |
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Inner product of 2 vectors |
Let a= a₁i + a₂j + a₃k, and b= b₁i + b₂j + b₃k a·b= a₁b₁ + a₂b₂ + a₃b₃ The inner product of two vectors is a scalar quantity. The inner product is sometimes denotes but it is the same as a·b |
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5 properties of the inner product |
1. a·a≥0 2. a·a= 0 if and only if a=0 3. αa · b= α(a·b), as well as, a · βb= β(a·b) 4. a · (b+c)= a·b + a·c, as well as, (a+b) · c= a·c +b·c 5. a·b = b·a |
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Length of a vector |
Let a= a₁i + a₂j + a₃k Length= √(a₁² + a₂² + a₃²) The length of a is denotes by ‖a‖, which is called the norm |
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The norm of a vector |
denoted ‖a‖, it is the quantity of the length of a ‖a‖= (a·a)^.5 |
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Unit vectors |
Vectors with norm 1 |
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Process to find normalized vector |
For any nonzero vector a, a/‖a‖ is a unit vector, and the result causes us to have normalized a |
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Distance between endpoints |
Let endpoints be denoted a,b ‖a-b‖ = distance |
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Distance between vectors |
Let the vectors be denotes P,Q ‖PQ‖ = distance |
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Angle between two vectors |
θ= cos⁻¹((a·b)/(‖a‖ ‖b‖)) |
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Corollary: Cauchy-Schwarz Inequality |
For any two vectors a and b, we have |a·b|≤ ‖a‖ ‖b‖ with equality if and only if a is a scalar multiple of b, or one of them is 0 |
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Orthogonal projection |
The orthogonal projection of v on a is the vector p= ((a·v)/‖a‖²)a |
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Triangle Inequality Theorem |
For vectors a and b in space, ‖a+b‖ ≤ ‖a‖ + ‖b‖ |
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Displacement and velocity |
If an object has a (constant) velocity vector v, then in t units of time the resulting displacement vector of the object is d=tv; thus, after time t=1, the displacement vector equals the velocity vector |
