Explanation

To find the acute angle between the vectors, first find the dot product:

z1 = 3 - 4i

z2 = -4 + 3i

Dot product = (3)(-4) + (-4)(3) = -12 - 12 = -24

Next, find the magnitudes:

|z1| = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5

|z2| = √((-4)^2 + 3^2) = √(16 + 9) = √25 = 5

Now, find the cosine of the angle:

cos(θ) = (Dot product) / (|z1| |z2|)

= -24 / (5 * 5)

= -24 / 25

θ = arccos(-24/25) ≈ 16°16'

Explanation

Explanation

Given the function f(x) = 3x - 2

This is a linear function with a slope of 3. Since it's linear and defined for all real numbers x, its range is all real numbers.

As x varies from -∞ to ∞, f(x) also varies from -∞ to ∞.

Explanation

To find the value of c that satisfies the Mean Value Theorem, we need to find the derivative of the function f(x) = x^2 + 2x + 1.

f'(x) = 2x + 2

The Mean Value Theorem states that there exists a point c in the interval [1, 2] such that:

f'(c) = (f(2) - f(1)) / (2 - 1)

First, let's calculate f(2) and f(1):

f(2) = 2^2 + 2(2) + 1 = 9

f(1) = 1^2 + 2(1) + 1 = 4

Now, we can calculate the slope:

(f(2) - f(1)) / (2 - 1) = (9 - 4) / 1 = 5

Now, we set f'(c) equal to 5:

2c + 2 = 5

Solving for c, we get:

2c = 3

c = 3/2

Since c = 3/2 is between 1 and 2, it satisfies the Mean Value Theorem.

Explanation

Explanation

If the correlation coefficient is 0, the two regression lines are perpendicular.

This is because a correlation coefficient of 0 indicates no linear relationship between the variables.

This causing the regression lines to be at right angles to each other.

Explanation

To form a differential equation from a given function containing arbitrary constants:

• If the function has 'n' arbitrary constants, differentiate the equation 'n' times with respect to the independent variable.
• This process removes the arbitrary constants, leaving a differential equation.

Explanation

1. Commutative Law: In an Abelian group, for all elements $a$ and $b$ in the group, the operation satisfies $a\cdot b=b\cdot a$.

2. Named after Niels Henrik Abel: The term "Abelian" is derived from the name of the mathematician Niels Henrik Abel.

Explanation

Explanation