Angle between vector and z-axis

• Syrus
In summary, the conversation is about a problem that asks for the angle between the normal vector to a given surface and the z-axis. The solution involves using the dot product-cosine relation and the vector (0,0,1) to find the angle. The conversation also mentions not posting images and typing out the problem instead.
Syrus
1. Homework Statement

I am looking at problem 2.2 pictured above.
I have solved all portions of the question except the last part, which asks for the angle between the normal vector to the surface and the z-axis.
I am aware that the normal vector is simply equal to the gradient of the surface given (the L.H.S. of the equation in the second line of the problem statement. In order to find the angle between this normal and the z-axis (for which I am using the vector (0,0,1), I am using the familiar dot product-cosine relation: a•b = |a| |b| cos(x)

The Attempt at a Solution

Syrus said:
View attachment 110018 1. Homework Statement

I am looking at problem 2.2 pictured above.
I have solved all portions of the question except the last part, which asks for the angle between the normal vector to the surface and the z-axis.
I am aware that the normal vector is simply equal to the gradient of the surface given (the L.H.S. of the equation in the second line of the problem statement. In order to find the angle between this normal and the z-axis (for which I am using the vector (0,0,1), I am using the familiar dot product-cosine relation: a•b = |a| |b| cos(x)

The Attempt at a Solution

Do not post images (especially do not post them sideways!). Just type out the problem; it is simple enough and does not need a lot of work.

Last edited:
Anyone care to provide a meaningful response?

Not me. I'm not willing to lie down to read it.

1. What is the angle between a vector and the z-axis?

The angle between a vector and the z-axis is the angle formed when the vector is projected onto the xy-plane and the z-axis. This angle is measured in a counterclockwise direction from the positive x-axis to the vector.

2. How do you calculate the angle between a vector and the z-axis?

To calculate the angle between a vector and the z-axis, you can use the dot product formula: cos(theta) = (v dot k) / (|v|*|k|), where v is the vector and k is the unit vector along the z-axis. The resulting angle theta is the angle between the vector and the z-axis.

3. Can the angle between a vector and the z-axis be negative?

Yes, the angle between a vector and the z-axis can be negative. This occurs when the vector is in the fourth quadrant of the xy-plane, resulting in an angle between 180 and 270 degrees.

4. What is the maximum possible angle between a vector and the z-axis?

The maximum possible angle between a vector and the z-axis is 90 degrees. This occurs when the vector is perpendicular to the xy-plane and lies in the positive or negative z-direction.

5. Why is the angle between a vector and the z-axis important in physics and engineering?

The angle between a vector and the z-axis is important in physics and engineering because it helps determine the orientation of the vector in three-dimensional space. This information is crucial in many applications, such as calculating forces, velocities, and accelerations in three-dimensional systems.

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