Vectors in R2 and R3

Question 1

directed line segments and vectors

Answer 1

for every directed line segment there is a vector in the same direction and length

ex) points P and Q

vector of PQ = Q - P

= (q₁+q₂) - (p₁+p₂)

Question 2

distance between vectors

Answer 2

the distance of 2 vectors is the distance of their end points

(doesn’t matter the subtraction order)

Question 3

length of vector

Answer 3

length of a vector (v) is the distance from the zero vector to the vector (v)

length of vector = d(u,v) = square root[(u₁-v₁)²+(u₂-v₂)²]

Question 4

unit vector

Answer 4

a vector is called a unit vector if it’s length is equal to 1

vector / magnitude of vector (length)

Question 5

scalar multiplication of vector

Answer 5

if c is scalar multiple

cv = (cv₁, cv₂, cv₂)

Question 6

vector addition

Answer 6

Question 7

vector subtraction

Answer 7

u - v = (u₁-v₁, u₂-v​₂)

Question 8

negative vector

Answer 8

negative vector (-v) is opposite and same length as vector (v)

Question 9

i, j, k notation

Answer 9

(1,0,0) = i vector

(0,1,0) = j vector

(0,0,1) = k vector

Question 10

collinear vector

Answer 10

2 vectors are called collinear if one is a scalar multiple of the other

ex) Cu = v

u and v are collinear

Question 11

how do you check for collinearity?

Answer 11

1. the vectors need to have the same zero components (they have any)
   ex) (0,1,2) and (0,2,4)
2. the components have the same ratio