Find the magnitude and direction of the vector AZ: A(8,8) and Z (1,-3)

Well, let's calculate the magnitude of vector AZ first. To do that, we need to find the distance between point A and point Z using the distance formula, which is the square root of the sum of the squares of the differences in each coordinate. So, the distance is:

√((1-8)^2 + (-3-8)^2) = √((-7)^2 + (-11)^2) = √(49 + 121) = √170

Therefore, the magnitude of vector AZ is √170.

Now, let's determine the direction of vector AZ. To find the direction, we can calculate the angle between the vector and the positive x-axis. Using trigonometry, we can find that:

θ = arctan((-3-8)/(1-8)) = arctan(-11/-7) ≈ arctan(1.5714)

The arctan of 1.5714 is approximately 57.64 degrees.

So, the magnitude of vector AZ is √170 and the direction is approximately 57.64 degrees counterclockwise from the positive x-axis.

But seriously, I hope this vector isn't going to a party because it seems to be heading in the wrong direction!

To find the magnitude and direction of the vector AZ, we first need to find the components of the vector.

The components of the vector AZ can be found by subtracting the coordinates of point A from the coordinates of point Z:

AZ = Z - A

AZ = (1, -3) - (8, 8)

AZ = (1 - 8, -3 - 8)

AZ = (-7, -11)

Now, we can find the magnitude of the vector AZ by using the formula:

Magnitude = sqrt(x^2 + y^2)

where x and y are the components of the vector.

Magnitude of AZ = sqrt((-7)^2 + (-11)^2)

Magnitude of AZ = sqrt(49 + 121)

Magnitude of AZ = sqrt(170)

Magnitude of AZ ≈ 13.038

The direction of the vector AZ can be found by using the formula:

Direction = atan(y / x)

where x and y are the components of the vector.

Direction of AZ = atan((-11) / (-7))

Direction of AZ = atan(11/7)

Direction of AZ ≈ 57.99 degrees

Therefore, the magnitude of the vector AZ is approximately 13.038 and the direction is approximately 57.99 degrees.

To find the magnitude and direction of the vector AZ, you can follow these steps:

Step 1: Determine the components of the vector AZ
The components of a vector are the differences in the x-coordinates and y-coordinates of the starting and ending points. Let's denote the coordinates of point A as (x₁, y₁) and the coordinates of point Z as (x₂, y₂).

Given A(8,8) and Z(1,-3), the x-coordinate difference is:
x₂ - x₁ = 1 - 8 = -7

The y-coordinate difference is:
y₂ - y₁ = -3 - 8 = -11

So, the components of the vector AZ are (-7, -11).

Step 2: Calculate the magnitude of the vector AZ
The magnitude of a vector can be found using the Pythagorean theorem. The formula for finding the magnitude (denoted as ||AZ||) is:

||AZ|| = √(x² + y²)

Using the components of AZ, we can substitute the values into the formula:
||AZ|| = √((-7)² + (-11)²)
= √(49 + 121)
= √170
≈ 13.038

Therefore, the magnitude of the vector AZ is approximately 13.038.

Step 3: Determine the direction of the vector AZ
The direction of a vector can be found by finding the angle it makes with the positive x-axis. You can use trigonometry to do this.

The direction θ (theta) of the vector AZ can be found using the following formula:

θ = tan^(-1)(y/x)

Substituting the values of the components of AZ into the formula:
θ = atan((-11)/(-7))

Using a calculator or a trigonometric table, you can find that:
θ ≈ 57.99°

Therefore, the direction of the vector AZ is approximately 57.99° (counterclockwise from the positive x-axis).

|AZ| = √(7^2 + 11^2) = √170

tanθ = -11/-7
θ = 57.5° but that's just a reference angle
for the vector from A to Z, θ = 237.5°