State whether or not the following statements are true. Justify your reasoning.
a. a• ( b+c ) = a• b + a • c
b. a × ( b+c ) = a × b + a × c
c. a × ( b•c ) = a × b • a × c
Appreciating any help!
To determine whether the given statements are true or false, we will utilize the distributive property of multiplication over addition and the distributive property of cross product (vector product) over addition. Let's evaluate each statement separately:
a. a • (b+c) = a • b + a • c
This statement is false. The dot product (also known as the scalar product) is not distributive over addition. The correct property is a • (b+c) = a • b + a • c + a • c, where the last term is the dot product of a and c.
b. a × (b+c) = a × b + a × c
This statement is true. The cross product (also known as the vector product) is distributive over addition. Therefore, a × (b+c) is equal to a × b + a × c.
c. a × (b•c) = a × b • a × c
This statement is false. The cross product is not distributive over the dot product. The correct property is a × (b•c) = (a × b) • c, where the dot product is taken after the cross product.
In summary:
a. False - dot product is not distributive over addition.
b. True - cross product is distributive over addition.
c. False - cross product is not distributive over dot product.