Verify using an example that Vector a + (Vector b • Vector c) is not equal to (Vector a + Vector b) • (Vector a +Vector c).

Explain the problem that arises

a+(b•c) is undefined. You cannot add a vector and a scalar.

Okay, imagine you have three vectors: Vector a, Vector b, and Vector c. Let's say Vector a = (2, 3), Vector b = (4, 5), and Vector c = (6, 7). Now, let's plug them into the equations to see what happens.

First, we have Vector a + (Vector b • Vector c).
So, we have (2, 3) + ((4, 5) • (6, 7)).
To calculate the dot product, we multiply corresponding elements and sum them up.
(4 * 6) + (5 * 7) = 24 + 35 = 59.
Therefore, Vector a + (Vector b • Vector c) equals (2, 3) + 59.

On the other hand, if we have (Vector a + Vector b) • (Vector a + Vector c),
we get ((2, 3) + (4, 5)) • ((2, 3) + (6, 7)).
Now, performing the addition operation, we get (6, 8) • (8, 10).
Again, calculating the dot product, we have (6 * 8) + (8 * 10) = 48 + 80 = 128.

So, here comes the problem: Vector a + (Vector b • Vector c) = (2, 3) + 59,
but (Vector a + Vector b) • (Vector a + Vector c) = (6, 8) • (8, 10) = 128.
Since 59 is not equal to 128, we can verify that these two expressions are not equal.

The issue arises due to the distributive property not applying to dot products.

To verify that Vector a + (Vector b • Vector c) is not equal to (Vector a + Vector b) • (Vector a + Vector c), let's consider the following example:

Let Vector a = (1, 2, 3)
Vector b = (4, 5, 6)
Vector c = (7, 8, 9)

Now, let's calculate both sides of the equation:

Vector a + (Vector b • Vector c) = Vector a + (Vector (b1*c1, b2*c2, b3*c3))
= Vector a + (4*7, 5*8, 6*9)
= Vector a + (28, 40, 54)
= (29, 42, 57)

(Vector a + Vector b) • (Vector a + Vector c) = (Vector (a1+b1, a2+b2, a3+b3) • Vector (a1+c1, a2+c2, a3+c3))
= (a1+b1)*(a1+c1) + (a2+b2)*(a2+c2) + (a3+b3)*(a3+c3)
= (1+4)*(1+7) + (2+5)*(2+8) + (3+6)*(3+9)
= 5*8 + 7*10 + 9*12
= 40 + 70 + 108
= 218

By comparing the results, we can see that (29, 42, 57) is not equal to 218. Hence, Vector a + (Vector b • Vector c) is not equal to (Vector a + Vector b) • (Vector a + Vector c).

The problem arises because the distributive property used in algebra does not hold for dot product operations. When expanding (Vector a + Vector b) • (Vector a + Vector c), it is incorrect to assume that it can be distributed into individual dot products.

To verify that Vector a + (Vector b • Vector c) is not equal to (Vector a + Vector b) • (Vector a +Vector c), we can consider a specific example.

Let's assume Vector a = [1, 2, 3], Vector b = [4, 5, 6], and Vector c = [7, 8, 9].

First, we will evaluate the left-hand side expression:

Vector a + (Vector b • Vector c) = Vector a + (Vector b dot product Vector c)
= [1, 2, 3] + ([4, 5, 6] • [7, 8, 9])
= [1, 2, 3] + (4 * 7 + 5 * 8 + 6 * 9)
= [1, 2, 3] + (28 + 40 + 54)
= [1, 2, 3] + 122
= [123, 124, 125]

Now, let's evaluate the right-hand side expression:

(Vector a + Vector b) • (Vector a + Vector c) = (Vector a + Vector b) dot product (Vector a + Vector c)
= ([1, 2, 3] + [4, 5, 6]) • ([1, 2, 3] + [7, 8, 9])
= ([1 + 4, 2 + 5, 3 + 6]) • ([1 + 7, 2 + 8, 3 + 9])
= ([5, 7, 9]) • ([8, 10, 12])
= 5 * 8 + 7 * 10 + 9 * 12
= 40 + 70 + 108
= 218

Therefore, Vector a + (Vector b • Vector c) is not equal to (Vector a + Vector b) • (Vector a +Vector c) in this example.

The problem that arises when comparing these two expressions is the difference in evaluation order. In the left-hand side expression, the dot product (Vector b • Vector c) is calculated first and then added to Vector a. On the right-hand side expression, the vectors Vector a and Vector b are added together first and then Vector a and Vector c are added together, and finally, the dot product is taken between these two resulting vectors.

These different evaluation orders can lead to different results. In general, the left-hand side expression represents the addition of Vector a with the dot product of Vector b and Vector c. Whereas, the right-hand side expression represents the dot product of the sum of Vector a and Vector b with the sum of Vector a and Vector c.

Hence, it is important to keep in mind the order of operations when working with vector operations to avoid confusion and mistakes.