Vectors Properties in Mathematical Calculations - Associative or Distributive?

Question

For cross products, the following property:

U X (V+W) = (U X V) + (U X W)
Distributive property

*Note*: X represents a multiplication sign.

For dot products:

U . (V+W) = U.V + U.W
Associative property

Why is it that these are regarded as different properties when essentially it is the same mathematical procedure?
Both should be regarded as distributive properties, shouldn't it?

We're not showing that associating groups of terms in any given set of order is true but that's what my book mentions. Why?

Answer 1

They're regarded differently because the first one refers to how the "cross product" is distributed to binary operations, and the other refers to how binary operations are associative under multiplication. Hence you can do the operations in whatever order you want for the dot product but you can't do the cross product in whatever order you want. Namely, the cross product isn't commutative; it is distributive. If you consider what your talking about instead the similarity of operations, you'll understand why this distinction is important. In addition, you're distributive property is wrong if X is multiplication because vectors can't be multiplied by other vectors.

Answer 2

subtraction is anti-commutative: a - b = -(b - a) (it truly is in many situations useful) there is the equality belongings of subtraction: if a = b, then a - c = b - c. 0 is a surprising-id for subtraction: a - 0 = a be conscious that there is no left-id for subtraction, this is to assert: some huge style x with x - a = a for each a (2a would not artwork for something yet a). subtraction does obey the distributive regulation: a(b - c) = ab - ac. it truly is likewise very useful. subtraction respects "multiples", if a is a dissimilar of ok, and b is a dissimilar of ok, so is a - b. in a manner subtraction is lots like <, it truly is "directional". in certainty, you are able to define < via: a < b if b - a is advantageous a = b if b - a is 0 a > b is a - b is advantageous. as a effect subtraction supplies us a thank you to inform which of two numbers is "greater". subtraction shows that advantageous numbers at the instant are not adequate to be waiting to clean up issues. with purely advantageous numbers, the equation: x + a million = 0 has no answer (if it did, then x = x + a million - a million might equivalent 0 - a million, which isn't advantageous).

Answer 3

subtraction is anti-commutative: a - b = -(b - a) (it particularly is often smart) there is the equality property of subtraction: if a = b, then a - c = b - c. 0 is a good-id for subtraction: a - 0 = a word that there's no left-id for subtraction, it particularly is to declare: some quantity x with x - a = a for each a (2a would not paintings for something yet a). subtraction does obey the distributive regulation: a(b - c) = ab - ac. it is likewise very smart. subtraction respects "multiples", if a is a dissimilar of ok, and b is a dissimilar of ok, so is a - b. in a fashion subtraction is lots like <, it quite is "directional". in fact, you may define < via: a < b if b - a is helpful a = b if b - a is 0 a > b is a - b is helpful. consequently subtraction provides us the thank you to tell which of two numbers is "larger". subtraction exhibits that helpful numbers are no longer sufficient to have the capacity to remedy issues. with in basic terms helpful numbers, the equation: x + a million = 0 has no answer (if it did, then x = x + a million - a million could equivalent 0 - a million, which isn't helpful).

Answer 4

yes, both are distributive properties

it must be a book error

however they are not essentially the same property

the dot product results in a scalar while the cross product results in a vector, they are very different things

Answer 5

I think this is due to what the dot product function is. say for

U = u1i + u2j, V=v1i+v2j, W= w1i + w2j.



U.(V+W) = (u1i + u2j) . (v1i + v2j + w1i + w2j) = (u1i + u2j) . ((v1+w1)i + (v2+w2)j) = u1(v1+w1) + u2(v2 + w2)



= u1v1 + u2v2 + u1w1 + u2w2 = U.V + U.W