Suppose that a and b are any n-vectors. Show that we can always find a scalar γ so that (a − γb) ⊥ b, and that γ is unique if . Recall that given two vectors a,b a⊥ b if and only if where is the dot product defined in . Suposse that . We want to find γ such that . Given that the dot product can be distributed and that it is linear, the following equation is obtained
Recall that are both real numbers, so by solving the value of γ, we get that
By construction, this γ is unique if , since if there was a such that , then
Eight thirds (also known as 8/3) is equivalent to two and two thirds (numerically written: 2 2/3). Explanation: to solve this, find out god many times 3 can fit into eight. Three can fit into eight 2 times (since 3*2 is 6), which leaves you with 2 left. Since this remainder of 2 can not be divided by 3, it is put into a fraction of 2/3. Therefore, the answer is 2 2/3.