Verify using an example that Vector a + (Vector b • Vector c) is not equal to (Vector a + Vector b) • (Vector a +Vector c). (This means that addition does not distribute over the dot product.) Explain the problem that arises.
If the vector A = 2.0i + 3.0j, vector B = 4.0j + 3.0k and vector C = 5.0i ‒5.0k, find the value of (A × B).C
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Let's verify that the expression "Vector a + (Vector b • Vector c)" is not equal to "(Vector a + Vector b) • (Vector a +Vector c)" using a specific example.
Consider the following vectors:
Vector a = [2, 3]
Vector b = [4, 5]
Vector c = [6, 7]
Now, let's calculate both expressions separately:
Expression 1: Vector a + (Vector b • Vector c)
First, we calculate the dot product of Vector b and Vector c:
Vector b • Vector c = (4 * 6) + (5 * 7) = 24 + 35 = 59
Next, we add Vector a to the dot product:
Vector a + (Vector b • Vector c) = [2, 3] + 59 = [2 + 59, 3 + 59] = [61, 62]
Expression 2: (Vector a + Vector b) • (Vector a + Vector c)
First, we calculate the sum of Vector a and Vector b:
Vector a + Vector b = [2, 3] + [4, 5] = [2 + 4, 3 + 5] = [6, 8]
Next, we calculate the sum of Vector a and Vector c:
Vector a + Vector c = [2, 3] + [6, 7] = [2 + 6, 3 + 7] = [8, 10]
Finally, we calculate the dot product of the resulting vectors:
(Vector a + Vector b) • (Vector a + Vector c) = [6, 8] • [8, 10] = (6 * 8) + (8 * 10) = 48 + 80 = 128
Now, comparing the results of both expressions:
Expression 1: Vector a + (Vector b • Vector c) = [61, 62]
Expression 2: (Vector a + Vector b) • (Vector a + Vector c) = 128
We can observe that the two expressions produce different results, indicating that they are not equal.
The problem that arises when attempting to distribute addition over the dot product is that the dot product is not distributive. This means that (Vector a • Vector b) + (Vector a • Vector c) is not equal to (Vector a • Vector b + Vector a • Vector c). In other words, the dot product does not follow the distributive property of addition.
To verify that Vector a + (Vector b • Vector c) is not equal to (Vector a + Vector b) • (Vector a +Vector c), we can use an example with actual vectors.
Let's say we have the following vectors:
Vector a = (1, 2, 3)
Vector b = (4, 5, 6)
Vector c = (7, 8, 9)
Now, let's calculate both sides of the equation:
1. Vector a + (Vector b • Vector c):
To calculate (Vector b • Vector c), we need to perform the dot product:
(Vector b • Vector c) = (4*7 + 5*8 + 6*9) = 94
Then, we can calculate Vector a + (Vector b • Vector c):
Vector a + (Vector b • Vector c) = (1, 2, 3) + 94 = (95, 96, 97)
2. (Vector a + Vector b) • (Vector a + Vector c):
First, we calculate the vectors (Vector a + Vector b) and (Vector a + Vector c):
(Vector a + Vector b) = (1, 2, 3) + (4, 5, 6) = (5, 7, 9)
(Vector a + Vector c) = (1, 2, 3) + (7, 8, 9) = (8, 10, 12)
Next, we calculate (Vector a + Vector b) • (Vector a + Vector c):
(Vector a + Vector b) • (Vector a + Vector c) = (5*8 + 7*10 + 9*12) = 5 + 70 + 108 = 183
Now, if we compare the results:
Vector a + (Vector b • Vector c) = (95, 96, 97)
(Vector a + Vector b) • (Vector a + Vector c) = 183
We can observe that the two results are not equal, indicating that Vector a + (Vector b • Vector c) is not the same as (Vector a + Vector b) • (Vector a +Vector c).
The problem that arises here is due to the difference between how scalar addition (represented by +) operates and how the dot product (represented by •) operates. Addition distributes over the dot product only when the dot product is treated as a scalar, not a vector.
In the equation (Vector a + Vector b) • (Vector a + Vector c), both (Vector a + Vector b) and (Vector a + Vector c) are treated as vectors, and the dot product is performed on these vectors. However, when we distribute the addition, we multiply each component separately instead of performing a dot product. This inconsistency leads to different results.