First 8 Weeks

Question 1

Projection Formula of b onto a

Answer 1

Question 2

Component of V onto U

Answer 2

Question 3

Dot Product

Answer 3

Question 4

What is the Cross Product? in 2-space and 3-space?

Answer 4

• The cross product in 3D is the line that’s orthogonal to 2 vectors and can be calulated by a × b = |a| |b| sin(θ) n or in the determinant form below
• The absolute value of the cross product is the volume of a parallelepiped in 3D
• The absolute value of the determinant of 2 vectors in 2D is the area of a parallelogram it is equal to the cross product of 2 vectors, but the cross product isn’t defined in 2d.

Question 5

Orthogonality - What’s point normal form?

Answer 5

The normal of the plane is the vector that’s orthogonal to the plane (a,b,c) and vector (a,b) for a line.

Question 6

Distance between a point and a line or a plane

Answer 6

Distance can be found by computing the rejection if given in parametric form or by the formula if given by a point-normal form

Question 7

The equation of a line that passes through x(0) and parallel to V

Answer 7

Question 8

Equation of a plane that is parallel to V1 and V2

Answer 8

Question 9

What’s the geometric representation of the cross product?

Answer 9

A = (base)(altitude) = |u||v| sin θ =

Question 10

When 3 vectors are on one plane?

Answer 10

When they all have an mutual initial point and the formula below is true

Question 11

The area of a parallelepiped

Answer 11

Question 12

the area of the parallelogram spanned by a=a1i+a2ja=a1i+a2j and b=b1i+b2jb=b1i+b2j is

Answer 12

Question 13

How to find the equation of the plane that passes through 3 points?

Answer 13

1. Find the vectors between the points that sit on the plane. (a1-a0)…
2. Find the cross product between 2 vectors
3. The cross product will be the and pick any point for the (x-x1)(y-y1)…

Question 14

What things do I need to describe a plane in a scalar equation?

Answer 14

1. A vector that lies in the plane
2. Normal vector to that vector from the plane

Question 15

How to write a linear combination from V1 & V2 to fit A or B

Answer 15

1. Take both vectors and ecuate to the new combination vecotor
2. Solve in a matrix to see if there are any number C1 or C2 that will give us the solutions.

Question 16

How to know if a linear combination spans {V1,V2}?

Answer 16

If it’s consistent and has solutions

Question 17

1. Find a vector equation and parametric equations of the line in R2 that passes through the origin and is parallel to the vector v = (−2, 3).
2. Find a vector equation and parametric equations of the line in R3 that passes through the point P0(1, 2,−3) and is parallel to the vector v = (4,−5, 1).

Answer 17

In 2D: Vector form: (x,y)=(0,0)+t(-2,3)
Parametric form: x=-2t, y=3t

In 3D: Vector form: (x,y,z)=(1,2,-3)+t(4,-5,1)
Parametric form: x=1+4t, y=2-5t, z=-3+t

Question 18

What 2 forms of vector equations we have to represent a 2-point vector line?

Answer 18

1. x=x0+t(x1-x0)
2. x=(1-t)x0+tx1