State whether or not the following statements are true. Justify your reasoning.?
a. Vector a • (Vector b + Vector c) = Vector a • Vector b + Vector a • Vector c
b. Vector a × (Vector b + Vector c) = Vector a × Vector b + Vector a × Vector c
c. Vector a × (Vector b • Vector c) = Vector a × Vector b • Vector a × Vector c
Any help is appreciated.
Q: What do you get when you cross a mosquito with a mountain climber?
A: Nothing -- you cannot cross a vector and a scaler.
I will just do two dimensions for demo
a)
A dot (B+C) = AxBx + Ax Cx + AyBy + AyCy
does that equal
AxBx+AyBy + AxCx+AyCy
Yes, it does
b)
A X (B+C) = A X (Bx+Cx)i +(By+Cy)j
i j k
Ax Ay 0
(Bx+Cx) (By+Cy) 0
find determinant
[Ax(By+Cy) - Ay(Bx+Cx) ]k
k direction of course
[ AxBy + AxCy - AyBx - AyCx ]k
is that the same as
A X B + A X C ???
[ AxBy-AyBx +AxCy-AyCx] k
YES, the same
c)
No way !!!
on left is a vector, on right is a scalar. in fact you can not even do the operation on the left which is a vector cross a scalar.
a. False. The dot product is distributive over addition, so Vector a • (Vector b + Vector c) is equal to (Vector a • Vector b) + (Vector a • Vector c).
b. False. The cross product does not distribute over addition, so Vector a × (Vector b + Vector c) is not equal to (Vector a × Vector b) + (Vector a × Vector c).
c. False. It is not possible to do a cross product on a dot product. So Vector a × (Vector b • Vector c) does not equal (Vector a × Vector b) • (Vector a × Vector c).
a. The statement is true.
The dot product (•) is distributive over vector addition. Therefore, vector a • (vector b + vector c) is equal to (vector a • vector b) + (vector a • vector c).
b. The statement is false.
The cross product (×) is not distributive over vector addition. Therefore, vector a × (vector b + vector c) is not equal to (vector a × vector b) + (vector a × vector c).
c. The statement is false.
The cross product (×) is not associative with the dot product (•). Therefore, vector a × (vector b • vector c) is not equal to (vector a × vector b) • (vector a × vector c).
To determine whether these statements are true or not, we need to understand the properties and operations of vectors. Let's go through each statement individually and justify our reasoning.
a. Vector a • (Vector b + Vector c) = Vector a • Vector b + Vector a • Vector c
The dot product, denoted as "•", of two vectors is distributive over addition, meaning we can distribute the dot product over the sum of two vectors. Therefore, the left side of the equation can be expanded as follows:
Vector a • (Vector b + Vector c) = (Vector a • Vector b) + (Vector a • Vector c)
The right side of the equation, Vector a • Vector b + Vector a • Vector c, is the sum of two dot products.
Since the left side and right side of the equation are the same expression, this statement is true.
b. Vector a × (Vector b + Vector c) = Vector a × Vector b + Vector a × Vector c
The cross product, denoted as "×", is not distributive over vector addition. In fact, the cross product is distributive over vector subtraction, but not addition. Therefore, the left side of the equation cannot be expanded as the sum of two cross products.
Thus, Vector a × (Vector b + Vector c) and Vector a × Vector b + Vector a × Vector c are not the same expression, meaning this statement is false.
c. Vector a × (Vector b • Vector c) = Vector a × Vector b • Vector a × Vector c
The cross product is not distributive over the dot product. In other words, we cannot distribute the cross product over the dot product.
Therefore, the left side of the equation, Vector a × (Vector b • Vector c), cannot be expanded as the product of two cross products.
Hence, Vector a × (Vector b • Vector c) and Vector a × Vector b • Vector a × Vector c are not the same expression, making this statement false.
In summary:
a. True
b. False
c. False