Problem2-AFor A=-I+2j +5k and B=2j-3k,find:

a) The dot productA.B
b) The angle between the A and B

To find the dot product of two vectors A and B, you need to take the sum of the products of their corresponding components.

a) Let's calculate the dot product A·B:
A = -i + 2j + 5k
B = 2j - 3k

To find the dot product, multiply the corresponding components and sum them up:

A·B = (-1)(0) + (2)(2) + (5)(-3)
= 0 + 4 - 15
= -11

Therefore, the dot product of vectors A and B is -11.

b) To find the angle between vectors A and B, you can use the formula:

cosθ = A·B / (|A| * |B|)

where θ is the angle between the two vectors and |A| and |B| represent the magnitudes of vectors A and B, respectively.

To find the magnitudes of the vectors:
|A| = sqrt((-1)^2 + (2)^2 + (5)^2)
= sqrt(1 + 4 + 25)
= sqrt(30)

|B| = sqrt((0)^2 + (2)^2 + (-3)^2)
= sqrt(4 + 9)
= sqrt(13)

Now substituting the values into the formula:

cosθ = -11 / (sqrt(30) * sqrt(13))

To find the angle θ, take the inverse cosine (arccos) of cosθ:

θ = arccos(-11 / (sqrt(30) * sqrt(13)))

Using a calculator, you can evaluate this expression to find the angle between vectors A and B.