Vector Geometry Unit 13

Question 1

Alpha

Answer 1

Angle made by function and X axis

Question 2

Beta

Answer 2

Angle made by function and Y axis

Question 3

Gamma

Answer 3

Angle made by function and Z axis.

Question 4

Relationship between alpha, beta, gamma, and vectors.

Answer 4

The unit vector of A =

COS(alpha)i + COS(Beta)j + COS(Gamma)k

Question 5

What makes two vectors orthogonal?

Answer 5

Their dot product is zero. Also, orthogonal means perpendicular.

Question 6

What formula can you use to solve for the angle measure between 2 vectors?

Answer 6

COS(theta) = (A DOT B)/(MAGNITUDE(A) * MAGNITUDE(B))

Question 7

What is a unit vector?

Answer 7

The components of the vector over the length of the vector.

Question 8

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

Answer 8

(A DOT B)/(Magnitude(A))

COMPa(b)

Question 9

What is the vector projection of B onto A?

Answer 9

It is the scalar projection of B onto A * the unit vector of A.

(A DOT B)/(Magnitude(A))^2 * A

PROJa(b)

Question 10

How can you calculate the work done by a force?

Answer 10

Work = Mag(F) * Mag(D) * COS(theta)

Work = F DOT D

The first formula is in magnitude space
The second formula is in vector space

Question 11

How do you calculate the resultant force of diagonal vectors?
What should you be careful off when doing this?
What if the vectors are not diagonal?
What space is this in?

Answer 11

1. a. Express the diagonal vectors by their components
   b. Sum the vectors
2. You should be careful of measuring theta from the same starting point. Always measure theta from the same quadrant on the X axis!
3. Then calculating the components of the vectors becomes easier!
4. This is in vector space! Direction is the most important element of these problems!

Question 12

When are two vectors parallel?

Answer 12

When the vectors have a scalar multiple or when their cross product compute to zero.

A scalar multiple is when each component of one vector multiplied by the same thing is equal to each corresponding component of another vector.

Question 13

When are two vectors orthogonal

Answer 13

When their dot product computes to zero.

Question 14

What is the relation of a cross product to the angle formed by its composite vectors?

Answer 14

CROSS(a,b) = MAG(A) * MAG(B) * SIN(theta)

Such that b is oriented vertically

Question 15

What is the relationship to vectors and a paralellogram?

Answer 15

Height = MAG(b) * SIN(theta)

Area = MAG(CROSS(a,b))

Such that b is oriented vertically

Question 16

What is the relationship of vectors and a parallelepiped?

Answer 16

Volume = A DOT MAG(CROSS(b,c))

Question 17

Torque?

Answer 17

Torque = MAG(R) * MAG(F) * Sin(theta)

Question 18

What is theta in torque system?

Answer 18

Angle formed by extended radius vector and force.

Question 19

Describe vector positions in torque system.

Answer 19

Torque vector and radius vector start at same point.

The force vector starts at the end of the radius vector.

Question 20

What is the relationship between dot products and cross products?

Answer 20

A cross product of two vectors will be orthogonal to both its composite vectors.

Or in math….

Cross(a,b) DOT a = 0
Cross(a,b) DOT b = 0