Multivariable Calculus Exam1

Front 1

What is the right hand rule?

Back 1

If thumb is z axis, if you close your open palm it goes from the x to the y axis.

Front 2

What is the distance formula in 3 dimentions?

Back 2

D = Sqrt[(x{2} - x{1})^2 + (y{2} - y{1})^2 + (z{2} - z{1})^2]

Front 3

What is the midpoint of two other points in three dimentions?

Back 3

Midpoint = ( ((x{1}+x{2})/2), ((y{1}+y{2})/2), ((z{1}+z{2})/2) )

Front 4

What is the equation of a sphere in R^3 with center at (h, k, j) and radius r ?

Back 4

(x-h)^2 + (y-k)^2 + (z-j)^2 = r^2

Front 5

What is a vector?

Back 5

A vector is a quantity that has both magnitude and direction.

Front 6

What is the zero vector?

Back 6

A vector whose lenghth is zero that points in any (and all) directions.

Front 7

How do you do vector addition and subtraction?

Back 7

Vector addition and subtraction are done by simply doing piecewise addition and subtraction of each of the vectors components two parts.

Front 8

How do you calculate a scalar times a vector?

Back 8

A scalar times a vector results in the same vector with each component of the vector multiplied by the scalar.

Front 9

What is a scalar?

Back 9

A scalar is not a vector. A scalar is simply a number like 4 or a variable like z.

Front 10

What is the vector projection of vector b onto vector a?

Back 10

proj(a)b

Front 11

How do you calculate the dot product of two vectors?

Back 11

given vector A and vector B:
A.B = a{1}*b{1} + a{2}*b{2} + a{3}*b{3}

Front 12

List the properties of dot products

Back 12

Given vector a, vector b, and vector c:
a.a = |a|^2
a.(b+c)=a.b+a.c
a.b=b.a
(given k scalar)k(a.b)=(ka).b=a.(kb)
a.(the zero vector) = (the number 0, not the vector 0, but the number 0)

Front 13

What is the geometric interpretation of the dot product?

Back 13

Given vector a and vector b:
a.b = |a|*|b|*cos(theta)

Front 14

What is the scalar projection of vector A onto vector B?

Back 14

Fill in later

Front 15

What is the Cross Product?

Back 15

The Cross Product of vector A and B is the resultant vector perpendicular to the plane formed by A and B in the direction of the right hand rule.

AXB = <a{2}b{3} - a{3}b{2}, a{3}b{1} - a{1}b{3}, a{1}b{2} - a{2}b{1}>

Front 16

List the properties of cross products?

Back 16

Given vector A, vector B, and vector C:
1)A X B = -(B X A)
2)A X A = 0
3)The vector A X B is orthogonal to both a and B.
-If vector v = A X B, then v.A = 0 & v.B = 0
4)A X (B+C) = A X B + A X C
5)A.(B X C) = scalar triple product = (a X B).C
6)A X (B X C) = vector triple product = (A.C)b-(a.b)C