when does vector (a+b) = (a-b) mean? explain

To understand when the equation vector (a+b) = (a-b) is true, let's first define what vectors and vector addition/subtraction mean.

In mathematics, a vector is a quantity that has both magnitude (length) and direction. It is commonly represented by an arrow. Vectors can be added or subtracted by performing operations on their components.

Now, let's break down the equation:

Vector (a+b) represents the vector resulting from adding vector a and vector b together.

Vector (a-b) represents the vector resulting from subtracting vector b from vector a.

For the equation to be true, it means that the vector resulting from adding vector a and vector b is equal to the vector resulting from subtracting vector b from vector a. This condition is satisfied when the two sides of the equation have the same magnitude and direction.

To determine when this occurs, we can consider the components of the vectors. Suppose vector a has components (a1, a2, a3) and vector b has components (b1, b2, b3). We can then write the equations for vector addition and subtraction:

(a+b) = (a1+b1, a2+b2, a3+b3)
(a-b) = (a1-b1, a2-b2, a3-b3)

Setting these two equal, we get the following equations:

a1+b1 = a1-b1
a2+b2 = a2-b2
a3+b3 = a3-b3

To simplify these equations further, notice that when we set the components equal to each other (a1 = -b1, a2 = -b2, a3 = -b3), we can conclude that a and b are equal in magnitude but opposite in direction.

So, the equation vector (a+b) = (a-b) is true when vector a and vector b have the same magnitude and opposite directions. In other words, when a is equal to -b.