Quizzes - Calculus 3

The distance |P₁P₂| between the points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂) is

|P₁P₂| =

sqrt((x2-x1)² + (y2-y1)² + (z2-z1)²)

The equation of a sphere with the center (a, b, c) and radius r:

r² =

(x-a)² + (y-b)² + (z-c)²

Given the points A(x₁, y₁, z₁) and B(x₂, y₂, z₂), the vector a with representation AB^→ is

a = <x2-x1, y2-y1, z2-z1>

Let a = <a₁, a₂, a₃> and b=<b₁, b₂, b₃> be vectors.

The length of a, |a|=

sqrt(a1² + a2² + a3²)

Let a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>

A unit vector in the same direction as a is u =

a/|a|

Let a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>

The dot product a⋅b =

a1b1 + a2b2+ a3b3

Let a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>

If the angle between a and b is θ(0 < θ < π), then we can write a⋅b=

|a||b|cos(theta)

Let a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>

The vectors a and b are orthogonal if

a⋅b = 0

Let a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>

The scalar projection of b onto a is given by:

compa(b) = (a⋅b)/|a|

Let a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>

The vector projection of b onto a is given by:

proja(b)=((a⋅b) / |a|)(a / |a|)

Let a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃>

The cross product of a and b is the vector

a x b = | i j k|=<a2b3-a3b2, a3b1-a1b3, a1b2-a2b1>

|a1 a2 a3|

|b1 b2 b3|

A vector orthogonal to a and b is

a x b

If θ is the angle between a and b (0 < θ < π), then (cross product)

|a x b| = |a||b|sin(theta)

An equation of the line through the point P(x₀, y₀, z₀) parallel to the vector v=<a, b, c>:

L(t)=<x₀+ta, y₀+tb, z₀+tc>

An equation of the plane through the point P(x₀, y₀, z₀) with normal vector n=<a, b, c>:

a(x-x₀)+b(y-y₀)+c(z-z₀)=0