Section 1

Question 1

A Dot Product returns a (Scalar/Vector)

Answer 1

Scalar

Question 2

A Dot Product is performed with

Answer 2

a1 * b1 + a2 * b2 + a3 * b3

Question 3

A Cross Product returns a (Scalar/Vector)

Answer 3

Vector

Question 4

A Cross Product is performed with

Answer 4

the determinant of a matrix using the vectors, evaluating to:
(a2b3 - a3b2)i - (a1b3 - a3b1)j + (a1b2 - a2b1)k

Question 5

(a + b) x c =

Answer 5

(a x c) + (b x c)

Question 6

a x b =

Answer 6

-b x a

Question 7

The area of a parallelopiped is

Answer 7

a * (b x c) |

Question 8

A parallelopiped is (definition)

Answer 8

A 3d object with 6 sides, each side being a parallelogram

Question 9

find the angle x between vectors a and b

Answer 9

a * b = ||a|| * ||b|| * cos(x)

Question 10

||a|| is the same as

Answer 10

sqrt(a * a)

Question 11

A Triple Product is

Answer 11

a * (b x c)

Question 12

A Triple Product returns a (Vector/Scalar)

Answer 12

Scalar

Question 13

a * ( b x c) is equivalent to

Answer 13

b * (a x c) and c * (a x b)

Question 14

Equation of a plane

Answer 14

the vector perpendicular to the plane (n) is expressed by (a, b, c). A point in the plane is (x0, y0, z0). The plane is then a(x - x0) + b(y - y0) + c(z - z0) = 0, simplified to ax + by + cz = D

Question 15

The distance between a plane and a point not on the plane

Answer 15

d = (Ax * n) / ||n|| where A is a point in the plane, x is the point off the plane, and n is a vector normal to the plane

Question 16

Equation of a vector function

Answer 16

(vector)x(t) = (x(t), y(t), z(t))

Question 17

Equation of a linear vector function

Answer 17

x(t) = a + tu where a and u are constant vectors.

Question 18

Equation of a circular vector function

Answer 18

x(t) = (r * cos(t), r * sin(t)) where r is the radius of the circle

Question 19

Finding Arc Length

Answer 19

If the curve x(t) is smooth, then the length of the curve between t = a and t = b is the integral from a to b of ||x’(t)|| with respect to t

Question 20

Tangent Vector

Answer 20

If x(t) is smooth, then x’(t) is tangent to x(t) at t.

Question 21

Notation for the Unit Tangent Vector

Answer 21

T(t)

Question 22

Notation for Curvature

Answer 22

K(t)

Question 23

Curvature

Answer 23

K(t) = (dT/dt) / ||x’(t)||

Question 24

Normal Vector

Answer 24

T’(t) / ||T’(t)||

Question 25

Indefinite Quadratic Form

Answer 25

4AC - B^2 < 0

Question 26

Semidefinite Quadratic Form

Answer 26

4AC - B^2 = 0

Question 27

Definite Quadratic Form

Answer 27

4AC - B^2 > 0

Question 28

Zero Set of a quadratic Form

Answer 28

Solve for Ax^2 + Bxy + Cy^2 = 0

Question 29

Tangent Planes

Answer 29

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