Vectors II

Question 1

What is the dot product when vectors are perpendicular?

Answer 1

0

Question 2

When the dot product is < 0, what is the angle range between the two vectors?

Answer 2

90° < θ < 180°

Question 3

When the dot product is > 0, what is the angle range between the two vectors?

Answer 3

0° ≤ θ < 90°

Question 4

What is the angle between vectors formula after rearranging for theta?

Answer 4

θ = cos-1 [ (a. b) / (|a| |b|)

Question 5

What is the angle between vectors formula before rearranging for theta?

Answer 5

(a. b) = |a||b|cosθ

Question 6

T or F: Vectors never exceed 180°

Answer 6

True :)

Question 7

What is a scalar projection?

Answer 7

A single number representing a length

Question 8

What is a vector projection?

Answer 8

A vector quantity corresponding to the vector colinear to b with magnitude equal to the scalar projection

Question 9

Scalar projection formula?

Answer 9

S.proj b a = (a . b) / |b|

Question 10

Vector projection formula?

Answer 10

V.proj b a = [(a . b) / |b|2] b

Question 11

Is cross product only used in R3? How can a R2 still be used?

Answer 11

Yes. Replace the z value with 0 if you want to cross product a line in R2

Question 12

What does the cross product find out?

Answer 12

The result is a vector that is perpendicular to the two vectors

Question 13

What is a plane?

Answer 13

An infinitely large flat 2d surface in R3

Question 14

x - 2y -3z = -4 represents…

Answer 14

A scalar/cartesian equation of a plane

Question 15

If the planes are perprendicular, the ____ are perpendicular

Answer 15

normals

Question 16

What are the possible types of intersections of a line and a plane? Describe each one.

Answer 16

Intersect @ 1 point - one solution, dot product ≠ 0.
Parallel & Distinct - zero solutions, dot product = 0 and no shared points
Coincident - infinite solutions, dot product = 0 and have shared points

Question 17

What is the cross product method? (Gabe’s way of memorizing, so ignore)

Answer 17

a2xb3 - opp
a3xb1 - opp
a1xb2 - opp

Question 18

What is the relationship between a scalar projection and a vector projection?

Answer 18

The magnitude of the vector projection is equal to the value of the scalar projection

Question 19

What is implemented into the scalar projection formula to create the vector projection formula?

Answer 19

b’s unit vector ( b/|b| ) is multiplied to the scalar projection formula

Question 20

What does it mean to Span R2?

Answer 20

Means that every vector in R2 can be expressed as a linear combination of vectors. Vectors span R2 if they are not colinear.

Question 21

What is a linear combination?

Answer 21

The sum of multiples of two or more vectors

Question 22

What are parametric equations?

Answer 22

Equations where x y and z (sometimes) are related to another variable (known as the parameter)

Question 23

How many normal vectors are there in R3?

Answer 23

Infinite

Question 24

If lines are colinear? What are the possible intersections? How do you determine between the two?

Answer 24

P&D - no solutions
Coincident - infinite solutions
Pick one point and sub it into both equations. If there there is an equivalent parameter value, lines are coincident

Question 25

If lines are non-colinear? What are the possible intersections? How do you determine between the two?

Answer 25

Skewed - No solution
Intersect @ 1 point - 1 solution
Make 1 = 2 and…