How Do You Calculate the Components of Vector C?

[princessfrost]

You are given vectors A= 4.7i - 7.0j and B= - 3.2i+ 6.9j . A third vector C lies in the xy-plane. Vector C is perpendicular to vector A and the scalar product of vector C with vector B is 16.0.

What is the x-component of vector C?

What is the y-component of vector C?




I'm not sure wither to use a dot product to solve this or use the cross product. I haven't yet learned how to use the cross product of two vectors so I don't think its that. How would i go about solving this problem? I thought about setting vector B equal to 16 but that didn't work out to well. Can someone please help me? Thank you!

 

[physicsgirlie26]

Well you know that the dot product of a and c has to equal zero and the dot product of a and b has to equal 16. So set up your equations and solve for x and y.

 

[m2003]

To solve this problem, we can use the dot product to find the x and y components of vector C. The dot product of two perpendicular vectors is equal to 0, so we can set the dot product of vector C and A equal to 0 and solve for the components of C.

The dot product of two vectors is calculated by multiplying the corresponding components and then adding them together. So for vector C and A, we have:

CxA = 0
(Cx * 4.7) + (Cy * (-7.0)) = 0

Since we know that the scalar product of vector C with vector B is 16.0, we can also set the dot product of C and B equal to 16.0:

CxB = 16.0
(Cx * (-3.2)) + (Cy * 6.9) = 16.0

We now have a system of two equations with two unknowns (Cx and Cy). We can solve this system using algebraic methods, such as substitution or elimination, to find the x and y components of vector C.

Alternatively, we can also use the geometric interpretation of the dot product to solve this problem. The dot product of two vectors is equal to the magnitude of one vector multiplied by the magnitude of the projection of the other vector onto the first vector. In this case, we know that the dot product of C and B is 16.0, so the magnitude of the projection of B onto C must be 16.0.

Using this information, we can draw a diagram and use trigonometry to find the x and y components of vector C. The x component can be found by taking the cosine of the angle between vector C and B, and multiplying it by the magnitude of B (which is 6.9). Similarly, the y component can be found by taking the sine of the angle and multiplying it by the magnitude of B.

I hope this helps you understand how to solve this problem. If you are still having trouble, I suggest seeking additional resources or asking your teacher for clarification. It's important to fully understand the concepts before moving on to more complex problems. Good luck!