Verifying properties of vector addition and scalar multiplication?

Question

I'm learning about vectors in trigonometry. Are there any websites that shows steps to verify these properties?

1. u + (v + w) = (u+ v) + w

2. 0 + v = v + 0 = v

3. v + (-v) = 0

4. u + v = v + u

5. a(u + v) = au +av

6. (a + b)v = av + bv

7. (ab)v = a(bv)

8. 1v = v

Answer 1

It's not really too hard to do this.



In each case, just write the vector as the sum of components and just do each operation component by component, showing that both sides end up the same.



I'm going to write the vectors as:







by which I mean:



xu i + yu j + zu k



Also:



* a and b are scalars

* u =

* v =

* w =

* 0 is the "zero vector" <0, 0, 0>

* 1 is the scalar (regular number) 1

* I use × to mean regular scalar multiplication, not "cross-product". (Don't worry if you don't know that term.)



I also say what I think each line would be called, but that's just my best guess. :c)



1) u + (v + w) = (u + v) + w (vector associative property of addition)



+ ( + ) = ( + ) +



+ = +



=



See how that worked? That was the hardest one.



2) 0 + v = v + 0 = v (vector identity property of addition)



<0, 0, 0> + = + <0, 0, 0> =



<0 + vx, 0 + vy, 0 + vz> = =



= =



3) v + (-v) = 0 (vector inverse property of addition)



+ - = <0, 0, 0>



+ <-vx, -vy, -vz> = <0, 0, 0>



= <0, 0, 0>



<0, 0, 0> = <0, 0, 0>



4) u + v = v + u (vector commutative law of addition)



+ = +



=



5) a(u + v) = au + av (distributive property of scalars over vector addition)



a( + ) = a + a



a() = +



=



=



ok, that one was a little messy....



6) (a + b)v = av + bv (distributive property of scalar addition over vectors)



(a + b) = a + b



<(a + b) × vx, (a + b) × vy, (a + b) × vz> = +



=



7) (ab)v = a(bv) (scalar-vector associative property of multiplication)



(a × b) = a(b)



= a()



=



9) 1v = v (scalar-vector identity property of multiplication)



1 =



<1 × vx, 1 × vy, 1 × vz> =



=

Answer 2

Scalar Addition

Answer 3

1. V is not a vector space, It fails the additive identity x + 0 = 0 + x = x , for all X in V, but x + 0 = x*0 = 0 which does not equal x

Answer 4

In order to understand the properties of Vector Addition, click on the link to Watch the VIDEO explanation:

http://bit.ly/1zn1A01



Properties Of Vector Addition

For any 2 vectors a and b

a plus b is equal to b plus a

Consider a parallelogram ABCD Let AB be equal to a and BC be equal to b.

Using triangle Law in triangle ABC we get AC is equal to a plus b since the opposite side of a parallelogram are equal and parallel. We get AD is equal to BC is equal to b and Dc is equal to AB is equal to a.

Again using triangle Law in triangle ABC we get AC is equal to b plus a Hence a plus b is equal to b plus a.

This is the commutative property of vector addition.

for any 3 vectors a, b and c

a plus b plus c is equal to a plus b plus c. let the vectors a ,b and c be represented by PQ, QR and RS respectively.

Using triangle Law in triangle PQR we get a plus b is equal to PQ plus QR is equal to PR

Using triangle Law in triangle PRS we get a plus b plus c is equal to PR plus RS is equal to PS

Using triangle Law in triangle QRS we get b plus c is equal to QR plus RS is equal to QS

Using triangle Law in triangle PQS we get a plus b plus c is equal to PQ plus QS equal to PS

Hence, a plus b plus c is equal to a plus b plus c.

This is the Associative property of vector addition.