To verify the equation Vector a + (Vector b • Vector c) = (Vector a • Vector b) + Vector c, let's break it down step by step and use an example.
First, let's define Vector a, Vector b, and Vector c. For simplicity, let's assume they are 2D vectors:
Vector a = (a1, a2)
Vector b = (b1, b2)
Vector c = (c1, c2)
Now, let's evaluate the left side of the equation: Vector a + (Vector b • Vector c).
The dot product (•) of Vector b and Vector c can be calculated as follows:
(Vector b • Vector c) = (b1 * c1) + (b2 * c2)
Now, let's calculate the result of the left side of the equation:
Vector a + (Vector b • Vector c) = (a1, a2) + [(b1 * c1) + (b2 * c2)]
Next, let's evaluate the right side of the equation: (Vector a • Vector b) + Vector c.
The dot product (•) of Vector a and Vector b can be calculated as follows:
(Vector a • Vector b) = (a1 * b1) + (a2 * b2)
Now, let's calculate the result of the right side of the equation:
(Vector a • Vector b) + Vector c = [(a1 * b1) + (a2 * b2)] + (c1, c2)
Now, let's compare the left and right sides of the equation:
(a1, a2) + [(b1 * c1) + (b2 * c2)] = [(a1 * b1) + (a2 * b2)] + (c1, c2)
If the equation holds true for any specific values of vectors a, b, and c, then it can be considered proven.
For example, let's consider the following values:
Vector a = (2, 3)
Vector b = (4, 5)
Vector c = (6, 7)
Substitute these values into the equation:
(2, 3) + [(4 * 6) + (5 * 7)] = [(2 * 4) + (3 * 5)] + (6, 7)
Simplifying both sides of the equation:
(2, 3) + [24 + 35] = [8 + 15] + (6, 7)
(2, 3) + 59 = 23 + (6, 7)
(2, 3) + (59) = (29) + (6, 7)
(61, 62) = (35, 36)
As the result on both sides of the equation is not equal, we can conclude that the equation Vector a + (Vector b • Vector c) = (Vector a • Vector b) + Vector c does not hold true for these values of vectors a, b, and c.
Therefore, the equation has been disproven using an example.
Please note that this is just one example, and to fully prove or disprove the equation, you would need to test it with various values of vectors a, b, and c.