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 need to use the fact that the dot product of two perpendicular vectors is zero. Given that a - 5b and a - b are perpendicular, we can write the equation:
(a - 5b) * (a - b) = 0
Expanding the dot product, we have:
a * a - a * b - 5b * a + 5b * b = 0
Since a and b are unit vectors, a * a = 1 and b * b = 1.
Substituting these values back into the equation, we get:
1 - a * b - 5b * a + 5 = 0
We can rearrange the equation to solve for a * b:
a * b + 5b * a = 6
Now, we factor out the common term b from the second and fourth terms:
(a + 5b) * b = 6
Since a and b are both unit vectors, the magnitude of a + 5b is |a + 5b| = √(1^2 + 5^2) = √26.
Therefore, we can simplify the equation to:
√26 * |b| = 6
Since b is a unit vector, |b| = 1.
So, we have:
√26 = 6
Squaring both sides:
26 = 36
This is not true, which means there is no value for a * b that satisfies the initial condition.
To determine the dot product of vectors a and b, we can use the property that if two vectors are perpendicular (i.e., their dot product is zero), then their dot product is equal to the negative product of their magnitudes:
(a - 5b) · (a - b) = 0
First, let's expand the dot product:
(a · a) - (a · b) - 5(a · b) + 5(b · b) = 0
Since a and b are unit vectors, their magnitudes are both equal to 1:
1 - (a · b) - 5(a · b) + 5 = 0
Rearranging the terms:
-4(a · b) = -6
Dividing both sides by -4:
(a · b) = -6 / -4 = 3 / 2
Therefore, the dot product of vectors a and b is 3/2.