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.