Chapter 3.1

Question 1

Parallellogram rule for vector addition

Answer 1

If v and w are vectors in 2-space or 3-space positioned so their initial point coincide, they then form two sides of a parallellogram, and the sum is the vector represented by the arrow from the common initial point to the opposite vertex of the parallellogram.

Question 2

Triangle rule for vector addition

Answer 2

If v and w are vectors in 2-space or 3-space that are positioned so the initial point of w is the terminal point of v. Then the sum v+w is represented by the arrow from the initial point of v to the terminal point of w.

Question 3

Vector addition viewed as translation

Answer 3

If two initial point coincide and the initial point of the sum of the vectors also coincide, then it’s terminal point can be found by translating one vector in the direction and length of the other vector. It’s terminal point will now be that of their sum.

accordingly, we say that v+w is the translation of v be w or, alternatively translation of w by v.

Question 4

Vector subtraction

Answer 4

The negative of a vector v, denoted -v is the vector that has the same length as v but is oppositely directed, and the difference of v from w, denoted w-v, is taken to be the sum:
w - v = w + (-v)

Question 5

Scalar multiplication

Answer 5

If v is a nonzero vector in 2-space or 3-space and if k is a nonzero scalar, then we define the scalar product of v by k to be the vector whose length is |k| times the length of v and whose direction is the same as that of v if k is positive and opposite to that of v if k is negative. If k = 0 or v = 0 then kv is defined to be 0.

Question 6

if v = (v(1), v(2), … v(n)) and w = (w(1), w(2), … w(n)) are vectors in R^(n), and if k is any scalar, then we define (4)

Answer 6

1) v+w = (v(1)+w(1), v(2)+w(2), … , v(n)+w(n))
2) kv = (kv(1), kv(2), … , kv(n))
3) -v = (-v(1), -v(2), … , -v(n))
4) w-v = w+(-v) = (w(1)-v(1), w(2)-v(2), … , w(n)-v(n))

Question 7

if u, v and w are vectors in R^(n), and if k and m are scalars then (8)

Answer 7

1) u + v = v + u
2) (u + v) + w = u + (v + w)
3) v + 0 = 0 + v = v
4) u + (-u) = 0
5) k(u + v) = ku + kv
6) u(k + m) = uk + um
7) k(mu) = (km)u
8) 1u = u

Question 8

Calculating vectors who’s initial point is not the origin

v = P(1)P(2)

Answer 8

v = (x(2) - x(1), y(2) - y(1))

Question 9

If v is a vector and k is a scalar, then (zero and negativ properties) (3)

Answer 9

1) 0v = 0
2) k0 = 0
3) (-1)v = -v

Question 10

linear combination of vectors

Answer 10

If w is a vector in R^(n), then w is said to be a linear combination of the vectors v(1), v(2), … v(n) in R^(n) if it can be expresssed in the form

w = k(1)v(1) + k(2)v(2) + … + k(n)*v(n), where k(1), k(2), … , k(n) are scalars. These scalars are called the coefficients of the linear combination in the case where r = 1 the formula becomes w = kv, so that a linear combination of a single vector is just a scalar multiple of that vector

Question 11

vector v and w are said to be equal (also called equivalent) in R^(n) if

Answer 11

v(1) = w(1), v(2) = w(2), … , v(n) = w(n)