Understanding Cross Product: Vector Product and Angle Separation Explained

In summary, the conversation discusses two problems. The first problem involves determining the result of the vector product x×(x×y) and the second problem involves finding the cross product of two vectors given their lengths and angle of separation. The conversation also includes a helpful explanation and visualization of the cross product for better understanding.
  • #1
theBEAST
364
0
I have two problems:


Homework Statement


In general, what can be said about the vector product x×(x×y)

The Attempt at a Solution


I thought the result of this would be parallel to y. However the answer suggests it is orthogonal to x. Can anyone explain how I could approach this question? I tried to visualize it in my head but it was very difficult.


Homework Statement


Given two vectors of length 2 and 3 separated by an angle of 30 degrees, what is the cross product of the two vectors?

The Attempt at a Solution


I know that the cross product = (length of vector a)*(length of vector b)*sin(theta)
This gives us 3.0, however the answer key suggests there is not enough information to answer the question.
 
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  • #2
theBEAST said:
I have two problems:

Homework Statement


In general, what can be said about the vector product x×(x×y)

The Attempt at a Solution


I thought the result of this would be parallel to y. However the answer suggests it is orthogonal to x. Can anyone explain how I could approach this question? I tried to visualize it in my head but it was very difficult.
You seem to be making an assumption about the relative orientations of ##\vec{x}## and ##\vec{y}##. Try look at a few specific cases and see if it clears up any misconceptions you have. Concentrate on the direction of the cross product. Don't worry about the magnitudes for now. Draw pictures!

First case, let ##\vec{x} = (1, 0, 0)## and ##\vec{y} = (0, 1, 0)##. That is the two vectors lie along the x-axis and y-axis respectively. What is ##\vec{x}\times\vec{y}##? And when you cross that result again with ##\vec{x}##, what do you get?

Second case, let ##\vec{x} = (1, 0, 0)## and ##\vec{y} = (1, 1, 0)##. This time, the second vector still lies in the xy-plane, but it's no longer aligned to the y-axis. Again, what is ##\vec{x}\times\vec{y}##? And when you cross that result again with ##\vec{x}##, what do you get? What effect did changing ##\vec{y}## have on the direction of the final answer?

Homework Statement


Given two vectors of length 2 and 3 separated by an angle of 30 degrees, what is the cross product of the two vectors?

The Attempt at a Solution


I know that the cross product = (length of vector a)*(length of vector b)*sin(theta)
This gives us 3.0, however the answer key suggests there is not enough information to answer the question.
This isn't quite right. The cross product gives you a vector, but quantity on the righthand side is a number. The two sides of your equation can't be equal. What you mean is
$$|\vec{a}\times\vec{b}| = |a||b|\sin\theta.$$ You found the magnitude of the cross product, but that's only half the answer. You still have to give its direction.
 
  • #3
vela said:
You seem to be making an assumption about the relative orientations of ##\vec{x}## and ##\vec{y}##. Try look at a few specific cases and see if it clears up any misconceptions you have. Concentrate on the direction of the cross product. Don't worry about the magnitudes for now. Draw pictures!

First case, let ##\vec{x} = (1, 0, 0)## and ##\vec{y} = (0, 1, 0)##. That is the two vectors lie along the x-axis and y-axis respectively. What is ##\vec{x}\times\vec{y}##? And when you cross that result again with ##\vec{x}##, what do you get?

Second case, let ##\vec{x} = (1, 0, 0)## and ##\vec{y} = (1, 1, 0)##. This time, the second vector still lies in the xy-plane, but it's no longer aligned to the y-axis. Again, what is ##\vec{x}\times\vec{y}##? And when you cross that result again with ##\vec{x}##, what do you get? What effect did changing ##\vec{y}## have on the direction of the final answer?



This isn't quite right. The cross product gives you a vector, but quantity on the righthand side is a number. The two sides of your equation can't be equal. What you mean is
$$|\vec{a}\times\vec{b}| = |a||b|\sin\theta.$$ You found the magnitude of the cross product, but that's only half the answer. You still have to give its direction.

Wow, thanks for clearing everything up! Makes sense now.
 

1. What is a cross product?

A cross product is a mathematical operation performed on two vectors to produce a third vector that is perpendicular to both of the original vectors. It is often used in physics and engineering to calculate torque, angular momentum, and magnetic fields.

2. How is a cross product calculated?

The cross product of two vectors, A and B, can be calculated using the following formula: A x B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k, where i, j, and k are the unit vectors in the x, y, and z directions respectively.

3. What is the difference between a dot product and a cross product?

A dot product is a mathematical operation that produces a scalar (a number) as a result, while a cross product produces a vector. In other words, a dot product calculates the projection of one vector onto another, while a cross product calculates the perpendicular component of two vectors.

4. What are the applications of cross products in real life?

Cross products have many applications in physics and engineering, including calculating the torque on a rotating object, determining the direction of magnetic fields, and solving problems involving angular momentum. They are also used in computer graphics to calculate lighting and shading effects.

5. Can a cross product be used with more than two vectors?

No, a cross product can only be performed on two vectors at a time. However, multiple cross products can be used together in a series to calculate the resultant vector of multiple vectors.

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