Vector A⃗ is in the direction 34.0 ∘ clockwise from the -y-axis. The x-component of A⃗ is Ax = -16.0 m .

What is the y-component of A⃗ ?

What is the magnitude of A⃗ ?

as always,

x = r cosθ -- use this to find r.
y = r sinθ

|A| = √(x^2+y^2) = r

To find the y-component of vector A⃗, we need to determine the length of the side that is opposite to the angle.

Given:
Direction of A⃗: 34.0° clockwise from the -y-axis
x-component of A⃗: Ax = -16.0 m

Let's use trigonometric functions to solve for the y-component.

In a right triangle, the cosine of an angle is defined as:
cos(θ) = adjacent side / hypotenuse

In this case, the adjacent side is the x-component (-16.0 m), and the hypotenuse is the magnitude of the vector A⃗. We can rewrite the formula as:
cos(θ) = Ax / A

where θ is the angle A⃗ makes with the -y-axis (34.0° clockwise). Rearranging the formula, we can solve for A:
A = Ax / cos(θ)

Now we can substitute the given values into the formula:
A = -16.0 m / cos(34.0°)

Using a calculator, we can calculate cos(34.0°) ≈ 0.829.

A ≈ -16.0 m / 0.829 ≈ -19.30 m

The y-component of vector A⃗ is approximately -19.30 m.

To find the magnitude of vector A⃗, we can use the Pythagorean theorem, which states that in a right triangle:
a² + b² = c²

In this case, a and b are the x-component and y-component of vector A⃗, and c is the magnitude of A⃗.

We have the values:
Ax = -16.0 m
Ay = -19.30 m

Using the formula, we can calculate:
A = sqrt(Ax² + Ay²)
A = sqrt((-16.0 m)² + (-19.30 m)²)

Using a calculator, we can simplify the calculation:
A ≈ sqrt(256 m² + 373.49 m²)
A ≈ sqrt(629.49 m²)
A ≈ 25.1 m

The magnitude of vector A⃗ is approximately 25.1 m.

To find the y-component of vector A⃗, we need to use trigonometry.

First, let's visualize the given information. We know that vector A⃗ is in the direction of 34.0∘ clockwise from the -y-axis. This means that if we draw a line in the direction of 34.0∘ clockwise from the -y-axis, vector A⃗ will lie on this line.

Next, we are given the x-component of A⃗, which is Ax = -16.0 m. This means that vector A⃗ has a length of 16.0 m in the negative x direction.

Now, let's use trigonometry to find the y-component of vector A⃗. Since vector A⃗ lies on the line that is 34.0∘ clockwise from the -y-axis, we can consider it as the hypotenuse of a right triangle. The x-component of A⃗ (-16.0 m) represents one side of the triangle, and the y-component represents the other side.

To find the y-component, we can use the formula:
y = hypotenuse * sin(angle)

In this case, the angle is the complement of 34.0∘, which is 90∘ - 34.0∘ = 56.0∘.

Substituting the values, we get:
y = 16.0 m * sin(56.0∘)

Using a calculator, we find that sin(56.0∘) is approximately 0.8290.

Therefore, the y-component of A⃗ is:
y = 16.0 m * 0.8290 ≈ -13.264 m (rounded to three decimal places)

Now, let's find the magnitude of vector A⃗. The magnitude can be found using the Pythagorean theorem, which states that the magnitude (or length) of a vector equals the square root of the sum of the squares of its components.

In this case, the magnitude of A⃗ can be found using the formula:
|A⃗| = sqrt(Ax^2 + Ay^2)

Substituting the values, we get:
|A⃗| = sqrt((-16.0 m)^2 + (-13.264 m)^2)

Using a calculator, we find that (-16.0 m)^2 + (-13.264 m)^2 = 471.998848 m^2

Therefore, the magnitude of A⃗ is:
|A⃗| ≈ sqrt(471.998848 m^2) ≈ 21.742 m (rounded to three decimal places)