Multivariable Calculus: Vectors in Space flashcards

Question 1

What is a vector?

Accepted Answer

A mathematical object with both magnitude and direction.

Question 2

What are equivalent vectors?

Accepted Answer

Vectors with the same magnitude and the same direction.

Question 3

What is a unit vector?

Accepted Answer

A vector whose magnitude is 1.

Question 4

How do you get the unit vector in the direction of a nonzero vector v?

Accepted Answer

Divide v by its magnitude: v/||v||.

Question 5

What is the magnitude of v = ⟨x1, y1, z1⟩?

Accepted Answer

||v|| = sqrt(x1^2 + y1^2 + z1^2)

Question 6

What is the standard equation of a sphere centered at (a, b, c) with radius r?

Accepted Answer

(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

Question 7

Into how many regions, called octants, do the coordinate planes divide space?

Accepted Answer

Eight.

Question 8

What is the dot product of u = ⟨u1, u2, u3⟩ and v = ⟨v1, v2, v3⟩?

Accepted Answer

u · v = u1v1 + u2v2 + u3v3

Question 9

How is the dot product u · v written using the angle θ between u and v?

Accepted Answer

u · v = ||u|| ||v|| cos θ

Question 10

What does v · v equal?

Accepted Answer

||v||^2, the square of the magnitude of v.

Question 11

What dot product makes vectors u and v orthogonal?

Accepted Answer

u · v = 0

Question 12

What are the direction cosines of a nonzero vector?

Accepted Answer

The cosines of the angles the vector makes with the coordinate axes.

Question 13

What is the vector projection of v onto u?

Accepted Answer

proj_u v = ((u · v)/||u||^2) u

Question 14

What is the scalar projection of v onto u?

Accepted Answer

comp_u v = (u · v)/||u||

Question 15

What is the work W done by a constant force F over a displacement s?

Accepted Answer

W = F · s = ||F|| ||s|| cos θ

Question 16

What is the magnitude of the cross product u × v, where θ is the angle between u and v?

Accepted Answer

||u × v|| = ||u|| ||v|| sin θ

Question 17

How is u × v oriented relative to u and v?

Accepted Answer

It is orthogonal to both u and v, with its direction given by the right-hand rule.

Question 18

How does v × u relate to u × v?

Accepted Answer

v × u = -(u × v)

Question 19

What is v × v for any vector v?

Accepted Answer

The zero vector.

Question 20

What is the area of a parallelogram with adjacent sides u and v?

Accepted Answer

||u × v||

Question 21

What is the volume of a parallelepiped whose adjacent edges are u, v and w?

Accepted Answer

V = |u · (v × w)|

Question 22

What does a zero triple scalar product u · (v × w) tell you about u, v and w?

Accepted Answer

The three vectors are coplanar (and coplanar vectors give a zero triple scalar product).

Question 23

What is the torque τ of a force F acting at position vector r from the axis?

Accepted Answer

τ = r × F

Question 24

What are the parametric equations of the line through (x0, y0, z0) with direction vector ⟨a, b, c⟩?

Accepted Answer

x = x0 + ta, y = y0 + tb, z = z0 + tc

Question 25

What are skew lines?

Accepted Answer

Two lines that are not parallel and do not intersect.

Question 26

What is the scalar equation of the plane through (x0, y0, z0) with normal vector n = ⟨a, b, c⟩?

Accepted Answer

a(x - x0) + b(y - y0) + c(z - z0) = 0

Question 27

What is the distance from the point (x0, y0, z0) to the plane ax + by + cz + d = 0?

Accepted Answer

D = |ax0 + by0 + cz0 + d| / sqrt(a^2 + b^2 + c^2)

Question 28

Which quadric surface has the equation x^2/a^2 + y^2/b^2 + z^2/c^2 = 1?

Accepted Answer

An ellipsoid.

Question 29

Which quadric surface has the equation x^2/a^2 + y^2/b^2 - z^2/c^2 = 1?

Accepted Answer

A hyperboloid of one sheet.

Question 30

Which quadric surface has the equation z = x^2/a^2 + y^2/b^2?

Accepted Answer

An elliptic paraboloid.

Question 31

How do you convert cylindrical coordinates (r, θ, z) to Cartesian coordinates?

Accepted Answer

x = r cos θ, y = r sin θ, z = z

Question 32

How do you convert spherical coordinates (ρ, θ, φ) to Cartesian coordinates?

Accepted Answer

x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ

Question 33

In spherical coordinates (ρ, θ, φ), what angle does φ measure, and what is its range?

Accepted Answer

The angle from the positive z-axis to the segment from the origin to the point, with 0 ≤ φ ≤ π.

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