Verify using an example that Vector a • (Vector b • Vector c) = (Vector a • Vector b) • Vector c is not true. Explain your reasoning both numerically and by using the definition of the dot product. I am very confused as to what this means?! Thanks for the assistance.
It wants you to substitute values (can be anything as far as I know) for each vector into that equation, then explain based on what you know of the definition of a dot product its properties.
For example:
<2,3> dot (<1,4> dot <3,3> = (<2,3> dot <1,4>) dot <3,3>
Work that out, and the two sides should be different answers. Then explain why that is...I'm guessing due to that multiplication property or something along those lines...
Awesome! Thanks, billy!
To show that the statement Vector a • (Vector b • Vector c) = (Vector a • Vector b) • Vector c is not true, we will use an example.
Let's consider the following vectors:
Vector a = (1, 2)
Vector b = (3, 4)
Vector c = (5, 6)
Now let's calculate both sides of the equation:
Left-hand side:
Vector a • (Vector b • Vector c)
= Vector a • (Vector b • Vector c)
= Vector a • (b1 * c1 + b2 * c2)
= Vector a • (3 * 5 + 4 * 6)
= Vector a • (15 + 24)
= Vector a • 39
= 1 * 39 + 2 * 39
= 39 + 78
= 117
Right-hand side:
(Vector a • Vector b) • Vector c
= (Vector a • Vector b) • Vector c
= (a1 * b1 + a2 * b2) • Vector c
= (1 * 3 + 2 * 4) • Vector c
= (3 + 8) • Vector c
= 11 • Vector c
= 11 * (5, 6)
= (55, 66)
So, the left-hand side of the equation evaluates to 117, while the right-hand side evaluates to (55, 66). Since they are not the same, we can conclude that the statement Vector a • (Vector b • Vector c) = (Vector a • Vector b) • Vector c is not true.
Now let's explain the reasoning using the definition of the dot product.
The dot product of two vectors a and b is given by a • b = a1 * b1 + a2 * b2 + ... + an * bn, where n is the number of dimensions.
According to the given equation, we have:
Vector a • (Vector b • Vector c) = (Vector a • Vector b) • Vector c
Breaking it down step by step, we can see that:
(Vector b • Vector c) is a scalar resulting from the dot product operation between Vector b and Vector c.
(Vector a • (Vector b • Vector c)) then takes this scalar and performs the dot product operation with Vector a.
On the other hand, (Vector a • Vector b) is another scalar resulting from the dot product operation between Vector a and Vector b.
((Vector a • Vector b) • Vector c) takes this scalar and performs the dot product operation with Vector c.
In general, the dot product is not associative, meaning that (a • b) • c is not necessarily equal to a • (b • c). Therefore, the equation is not always true.
In the given example, we have shown numerically that the left-hand side (117) is not equal to the right-hand side ((55, 66)). This confirms the lack of equality between the two sides of the equation and supports the conclusion that the equation does not hold true in general.
To verify that the equation Vector a • (Vector b • Vector c) = (Vector a • Vector b) • Vector c is not always true, we can use a numerical example.
Let's consider the following vectors: Vector a = [1, 2, 3], Vector b = [4, 5, 6], and Vector c = [7, 8, 9].
Using the dot product definition, we can calculate each side of the equation:
1. Vector a • (Vector b • Vector c):
First, we compute the inner dot product of Vector b and Vector c: Vector b • Vector c = 4*7 + 5*8 + 6*9 = 94.
Then, we take this result and perform the dot product with Vector a: Vector a • (Vector b • Vector c) = 1*94 + 2*94 + 3*94 = 282 + 188 + 94 = 564.
2. (Vector a • Vector b) • Vector c:
We start by calculating the dot product of Vector a and Vector b: Vector a • Vector b = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Then, we multiply this result with Vector c: (Vector a • Vector b) • Vector c = 32 * [7, 8, 9] = [32*7, 32*8, 32*9] = [224, 256, 288].
Comparing the two results, we find that Vector a • (Vector b • Vector c) = 564, while (Vector a • Vector b) • Vector c = [224, 256, 288]. Therefore, the equation does not hold true for the given example.
To understand why this is the case, let's analyze the definition of the dot product. The dot product of two vectors a and b is calculated by taking the sum of the products of their corresponding components. In other words, it measures the projection of one vector onto the other.
In the equation Vector a • (Vector b • Vector c) = (Vector a • Vector b) • Vector c, we have two dot products being performed. The left side performs the dot product of Vector b and Vector c first, giving a scalar value. This scalar is then used to perform the dot product with Vector a.
On the right side, the dot product of Vector a and Vector b is calculated first, resulting in a scalar value. This scalar is then multiplied by Vector c, giving a vector.
The equation would only be true if the scalar value obtained on each side could be freely interchanged with the vector. However, in general, a scalar value cannot be treated the same as a vector when it comes to multiplication.
Therefore, the equation Vector a • (Vector b • Vector c) = (Vector a • Vector b) • Vector c is not always true.