Calculating the Product of Vectors A, B and C

In summary, the vectors A, B and C have components Ax = 3, Ay = -2, Az = 2, Bx = 0, By = 0, Bz = 4, Cx = 2, Cy = -3, Cz = 0. TheAttempt at a Solutionhey hypsm - where are you stuck?evaluates B+C first, but if you want to keep going your way it should beA X (B + C)=A X B+ AX C = n|A|B||sin(x) + m|A||C|sin(y).one good way to find the angel is using the dot product..
  • #1
hpysm
2
0

Homework Statement



The vectors A, B and C have components Ax = 3, Ay = -2, Az = 2, Bx = 0, By = 0, Bz = 4, Cx = 2, Cy = -3, Cz = 0. Calculate the A X (B + C) ??




Homework Equations





The Attempt at a Solution

 
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  • #2
hey hypsm - where are you stuck?

evaluate B+C first
 
  • #3
I am ding this way: A X (B + C)=AXB+AXC = AXBsin(x) + AXCsin(x); how I can get angle of x degree ?
 
  • #4
Don't you know how to calculate a cross product using the components of the two vectors? BTW, lanedance's suggestion will save you some calculating, since the addition of two vectors first is much simpler than taking two cross products and then adding.
 
  • #5
hpysm said:
I am ding this way: A X (B + C)=AXB+AXC = AXBsin(x) + AXCsin(x); how I can get angle of x degree ?

that doesn't quite make sense... AXB is a vector and the magnitude is given by
|AXB| = |A|.|B|sin(x).n

where n is a unit vector perpendicular to both A & B, and x is the angle between them.

the angle between A and B, will not in general be the same as that between A and C, neither will thr normal vector.

Hence I would do it the way first suggested... (evaluate B+C first)

however, if you want to keep going your way it should be
A X (B + C)=A X B+ AX C = n|A|B||sin(x) + m|A||C|sin(y)

one good way to find the angel is using the dot product..

if you were just looking to find the magnitude of the vector this would all simplify
 
  • #6
If [itex]\vec{A}= A_x\vec{i}+ A_y\vec{j}+ A_z\vec{k}[/itex] and [itex]\vec{B}= B_x\vec{i}+ B_y\vec{j}+ B_z\vec{k}[/itex] then, symbolically,
[tex]\vec{A}\times\vec{B}= \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{array}\right|[/tex]

You really need to know that before you can do this problem.
 

FAQ: Calculating the Product of Vectors A, B and C

1. How do I calculate the product of three vectors A, B and C?

To calculate the product of three vectors A, B, and C, you need to first find the cross product of two of the vectors, let's say A and B. Then, take that cross product and find the dot product with the third vector C. This will give you the final product of all three vectors.

2. What is the difference between the dot product and cross product of vectors?

The dot product of two vectors is a scalar quantity that represents the projection of one vector onto the other. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. The cross product, on the other hand, is a vector quantity that is perpendicular to both of the original vectors and has a magnitude equal to the area of the parallelogram formed by the two vectors. It is calculated by taking the determinant of a 3x3 matrix formed by the two vectors.

3. Can the product of vectors be negative?

Yes, the product of vectors can be negative. This depends on the angle between the two vectors and the direction of the resulting vector. The dot product can be negative if the angle between the two vectors is greater than 90 degrees, while the cross product can be negative if the resulting vector is pointing in the opposite direction of the right-hand rule.

4. What are some real-life applications of calculating the product of vectors?

Calculating the product of vectors has many real-life applications, such as in physics for calculating work, torque, and angular momentum. It is also used in engineering for calculating forces and moments in structures. In computer graphics and animation, the cross product is used to calculate the direction of reflected light and to create 3D models. It is also used in navigation and robotics for calculating position and orientation of objects.

5. What are some common mistakes when calculating the product of vectors?

One common mistake when calculating the product of vectors is forgetting to take the magnitude of the resulting vector in the cross product. Another mistake is using the wrong formula for the dot product or cross product, as they have different equations. Additionally, forgetting to consider the direction of the resulting vector can lead to incorrect calculations. It's also important to make sure that the two vectors being used are in the same coordinate system when calculating their product.

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