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

To determine the value of a * b, we can use the given information that vectors a - 5b and a - b are perpendicular.

Since two vectors are perpendicular if and only if their dot product is zero, we can set up the following equation:

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

Expanding the dot product, we get:

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

Since a and b are unit vectors, their magnitudes are both equal to 1, so a * a = 1 and b * b = 1. Additionally, since a * b is the dot product of unit vectors a and b, the dot product is also equal to the cosine of the angle between them.

Substituting these values into the equation above, we get:

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

Simplifying further:

-4(a * b) - 5(b * a) + 6 = 0

Since the dot product is commutative, (a * b) = (b * a), so we can substitute (a * b) for (b * a) in the equation:

-4(a * b) - 5(a * b) + 6 = 0

Combining like terms:

-9(a * b) + 6 = 0

Rearranging the equation:

-9(a * b) = -6

Finally, solving for (a * b):

(a * b) = -6 / -9

(a * b) = 2/3

Therefore, the value of a * b is 2/3.

To determine the dot product of the vectors a and b, we can use the property that perpendicular vectors have a dot product of 0.

Given that (a - 5b) and (a - b) are perpendicular, we can set up the dot product equation:

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

Expanding the dot product:

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

Since a and b are unit vectors, we know that (a * a) = 1 and (b * b) = 1:

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

Rearranging the terms:

-4(a * b) - 5(b * a) + 6 = 0

Since (a * b) is the same as (b * a), we can simplify the equation:

-9(a * b) + 6 = 0

Now, isolate (a * b) by moving the constant term to the other side:

-9(a * b) = -6

Divide both sides of the equation by -9:

(a * b) = -6 / -9

Simplifying the fraction:

(a * b) = 2/3

Therefore, the dot product of a and b is 2/3.