9: Vectors

Question 1

What is the vector form of a line?

Answer 1

r=a+λb

Question 2

What is the other vector form of a line?

Answer 2

r+c+λ(d-c)

Question 3

What is the Cartesian form of a 3D line?

Answer 3

x-a1/b1 = y-a2/b2 = z-a3/b3

Question 4

What is the vector equation of a plane?
And what does each letter represent?

Answer 4

r=a+λb+μc
a is the position vector of a point.
b and c are vectors in the plane.
λ and μ are scalars.

Question 5

Cartesian equation of a plane in 3 dimensions?

Answer 5

ax+by+cz=d
Remember vector(abc) is the normal vector to the plane!!!

Question 6

Scalar product of two vectors formula (a,b)?

Answer 6

a.b = |a||b|cosθ

Question 7

What 2 things should you know about dot products?

Answer 7

1. The non-zero vectors a and b are perp. if and only if a.b=0
2. If a and b are parallel, a.b = |a||b|
   In particular a.a = |a|^2

Question 8

How do you get the dot product from 2 cartesian form vectors?

Answer 8

If a=a1i + a2j + a3k and b-…
then a.b = vec(a1 a2 a3).vec(b1 b2 b3) = a1b1 + a2b2 + a3b3

Question 9

Formula for finding the acute angle between 2 intersecting line?

Answer 9

cosθ = |a.b/|a||b||

Question 10

What is the scalar product form of the equation of a plane?

Answer 10

r.n=k where k=a.n for any point in the plane with position vector a.

Question 11

Formula for finding the actue angle between line r=a+λb and plane r.n=k?

Answer 11

sinθ = | b.n/|b||n| |

Question 12

Formula for finding the actue angle between plane r.n1=k1 and plane r.n2=k2?

Answer 12

cosθ = |n1.n2/|n1||n2||