uzman1243 said:I was studying the dot product, and it says that (a.b).c makes no sense.
so if you do (a.b) can = to β
and then is it not possible to do β.c?
WHY can't you 'dot' a scalar and a vector? why?
Mark44 said:Because the dot product is defined ONLY for two vectors. You can multiply a vector by a scalar, and this is called scalar multiplication.
uzman1243 said:But why? is there any proofs as to why this is defined this way?
Matterwave said:The dot product, in two dimensions (for simplicity) is defined as:
$$\vec{a}\cdot \vec{b}=a_xb_x+a_yb_y$$
Now, this assumes ##\vec{a}=(a_x,a_y)## and ##\vec{b}=(b_x,b_y)## are vectors. What would it mean to turn ##a## into a number? Certainly you can "define" the "dot product" of a scalar and a vector as:
$$a\cdot\vec{b}=a\vec{b}=(ab_x,ab_y)$$
But that's just the same as a scalar product, so it would be supremely confusing to also call it a "dot product". That's why we don't call that the "dot product".
A definition doesn't have to be proved.uzman1243 said:But why? is there any proofs as to why this is defined this way?
ModusPwnd said:Can you prove that a cat is not a soda can?
This statement is not mathematically valid. In order to perform the operation of dot product, the two vectors must have the same number of components. In this case, (a.b) is a scalar and cannot be multiplied by c, which is a vector. Therefore, the statement does not make sense.
Yes, (a.b).c can be rewritten as a(b.c) or c(a.b). This is known as the associative property of dot product.
The dot product is a mathematical operation used to calculate the projection of one vector onto another. It is also used in physics to calculate work and in geometry to determine the angle between two vectors.
In some cases, (a.b).c may make sense if c is a scalar. This is because in scalar multiplication, the scalar value is multiplied by each component of the vector. Therefore, the statement (a.b).c can be interpreted as a scalar multiplied by a vector, which is valid.
Yes, (a.b).c can equal 0 if one of the vectors is a zero vector or if the angle between the two vectors is 90 degrees. This is because the dot product of two perpendicular vectors is always 0.