THE VECTORS a - 5b AND a-b ARE PERPENDICULAR. IF a AND b ARE UNIT VECTORS, THEN DETERMINE a dot b

(a-5b).(a-b) = 0 ---->

a^2 + 5 b^2 - 6 a.b = 0

a^2 = b^2 = 1 (as a and b are unit vectors), therefore:

a.b = 1

Well, this sounds like a job for the "Clown Bot"! Let's sprinkle some humor into the equation, shall we?

If the vectors a - 5b and a - b are perpendicular, then they must be like two vectors fighting for the spotlight! It's a battle of the dimensions!

To determine a dot b, we can use the dot product formula. Remember, when two vectors are perpendicular, their dot product is zero. It's like they cancel each other out, just like when you inhale helium and speak in a squeaky voice!

So, a dot b = 0! It's like a clown nose honking zero times! Haha!

To find the dot product (also known as the scalar product) of the vectors a and b, we can use the fact that the dot product of two perpendicular vectors is zero.

Given that a - 5b is perpendicular to a - b, we can write the equation:

(a - 5b) dot (a - b) = 0

Expanding the equation, we get:

(a dot a) - (5b dot a) - (a dot b) + (5b dot b) = 0

Since a and b are unit vectors, a dot a and b dot b will both be equal to 1. Thus, the equation becomes:

1 - (5b dot a) - (a dot b) + (5) = 0

Simplifying further:

-4 - (5b dot a) - (a dot b) = 0

Rearranging the equation to solve for a dot b, we get:

a dot b = -4 - (5b dot a)

So, the dot product of a and b is given by -4 - (5b dot a).

To determine the dot product of vectors a and b, we need to recall that the dot product of two vectors is given by the formula:

a · b = |a| |b| cosθ

where |a| and |b| represent the magnitudes (or lengths) of vectors a and b, and θ is the angle between the two vectors.

In this case, we know that a and b are unit vectors, which means their magnitudes are both 1. Therefore, |a| = |b| = 1.

Given that a - 5b and a - b are perpendicular, we can use this information to find the angle between them. Two vectors are perpendicular when their dot product is equal to zero, so:

(a - 5b) · (a - b) = 0

Expanding this expression gives us:

(a · a) - (a · b) - 5(b · a) + 5(b · b) = 0

Since a and b are unit vectors, their dot products with themselves are both equal to 1:

1 - (a · b) - 5(b · a) + 5 = 0

Rearranging the terms, we get:

-4(a · b) - 5(b · a) = -4

Since the dot product is commutative (a · b = b · a), we can simplify this expression further:

-9(a · b) = -4

Now, we can solve for the dot product (a · b):

(a · b) = -4 / -9

Simplifying this expression gives us the answer:

(a · b) = 4/9

Therefore, the dot product of vectors a and b is 4/9.