Dot/Cross product vector problem.

In summary, the homework equation is A=7.92, B=8.28, and C=9.68. The scalar/dot product of C and B is 13. The angle between B and C is 80.67 degrees.
  • #1
QuarkCharmer
1,051
3

Homework Statement


2zyd9qh.jpg


Homework Equations



The Attempt at a Solution


Well, I have the whole thing drawn out, and calculated some information that will be needed.

[itex]|A|=7.92[/itex]
[itex]|B|=8.28[/itex]

I am assuming that the question means that vector C is perpendicular with vector A, meaning, there is 90 degrees between them.

a.) Find the x component of C

It says that the scalar product of C and B is 13, so I am guessing that means I have to know the angle between those two vectors. Which would be the angle between B and A minus 90 degrees right?

How do I find the angle between B and A ?

Alright, so:
[tex]A\bullet B = (4.8)(-3.9)+(-6.3)(7.3)[/tex]
[tex]A\bullet B = -64.71[/tex]
and since [itex]A\bullet B = |A||B|cos(\theta)[/itex]
[tex]|A||B|cos(\theta) = -64.71[/tex]

[tex]cos(\theta) = \frac{-64.71}{(7.92)(8.28)}[/tex]
[tex]\theta = 170.67 degrees[/tex]

Then the angle between B and C is 80.67 degrees, which looks about right.

So then, the scalar/dot product of C and B is:
[tex]C \bullet B = |8.28||C|cos(80.67) = 13[/tex]
[tex]|C| = \frac{13}{cos(80.67)|8.28|}[/tex]
[tex]|C| = 9.68[/tex]Now what?
I don't see how the components of C are coming out of this.
 
Last edited:
Physics news on Phys.org
  • #2
The fact that the scalar product of C and B is 13 also means that CxBx + CyBy = 13 (by the definition of scalar product). Does that help at all?
 
  • #3
cepheid said:
The fact that the scalar product of C and B is 13 also means that CxBx + CyBy = 13 (by the definition of scalar product). Does that help at all?

I was thinking of that but then I would still be stuck with two unknowns.
[tex]C_{x}(-3.9)+C_{y}(7.3)=13[/tex]

but you probably said that for a reason (thanks) so I will look for a second system perhaps.
 
  • #4
QuarkCharmer said:
I was thinking of that but then I would still be stuck with two unknowns.
[tex]C_{x}(-3.9)+C_{y}(7.3)=13[/tex]

but you probably said that for a reason (thanks) so I will look for a second system perhaps.

Right, but you also know that C is perpendicular to A, which should give you a second equation involving Cx and Cy. Two equations and two unknowns ==> an exact solution can be found.
 
  • #5
Hmm,

I found the angle from the positive x-axis to A to be -52.67 degrees (used arctan on vector A's components).

If the angle between C and A is 90, then the angle between C and the positive x-axis must be 90-|-52.67| = 37.33

Since I know |C| = 9.68 I could just get the x/y components of vector C right there?

I get:
[tex]C_{x} = 7.69[/tex]
[tex]C_{y} = 5.87[/tex]
Does that seem right? (Edit: No, it's not right at all, ugh. Looking into cepheid's solution)
 
Last edited:
  • #6
If two vectors are perpendicular, then their dot product is zero. You can see this just by noting that cos(90°) = 0, but it's also helpful to understand the geometric argument. Recall that when you're taking the dot product of A and B, you sort of "project" A onto B by drawing a line starting from the tip of A that is perpendicular to A and extend it until it lands on B. This marks out the component of A that is in the same direction as B. So the dot product can be interpreted as the magnitude of B multiplied by the component of A that is in the direction of B.

However, if the two vectors are perpendicular, then there is NO component of A that is in the direction of B. When you draw a line from the tip of A perpendicular to A, it never intersects B.

In any case, the dot product being zero gives you your second equation involving the x and y components.
 
  • #7
Ah that's right. Sorry, I'm kind of teaching myself ahead of the course some).

My two systems of equations are:

-3.9x + 7.3y = 13
4.8x - 6.3y = 0

x=-1.3739
y= 1.047

Which both to two significant digits, is still incorrect?
x=-1.4
y= 1.1
 
  • #8
QuarkCharmer said:
Ah that's right. Sorry, I'm kind of teaching myself ahead of the course some).

My two systems of equations are:

-3.9x + 7.3y = 13
4.8x - 6.3y = 0

x=-1.3739
y= 1.047

Which both to two significant digits, is still incorrect?
x=-1.4
y= 1.1

Yeah, I get different answers. My strategy was to use the lower equation to solve for x in terms of y, and plug that expression for x into the upper equation in order to solve for y. Once you have y, you have x.
 
  • #9
Yes, definitely made a mistake on the system of equations.

The proper answer was x=7.8 and y=6.0 (to 2 s.f.)

I sure made that one more complicated than it needed to be. Thanks for the help, I appreciate it.
 
  • #10
QuarkCharmer said:
Yes, definitely made a mistake on the system of equations.

The proper answer was x=7.8 and y=6.0 (to 2 s.f.)

I sure made that one more complicated than it needed to be. Thanks for the help, I appreciate it.

You're welcome! :smile:
 

1. What is the dot product of two vectors?

The dot product of two vectors is a mathematical operation that results in a scalar quantity. It is calculated by multiplying the corresponding components of two vectors and then adding the products together.

2. How do you calculate the dot product of two vectors?

To calculate the dot product, you multiply the x-components of the vectors, then the y-components, and finally the z-components. Then, add all of these products together to get the final scalar value.

3. What does the dot product represent?

The dot product represents the similarity between two vectors. It is also equal to the product of the magnitudes of the vectors and the cosine of the angle between them.

4. What is the cross product of two vectors?

The cross product of two vectors is a mathematical operation that results in a vector quantity. It is calculated by taking the determinant of a 3x3 matrix formed by the two vectors and the unit vectors i, j, and k, and then simplifying the result.

5. How do you calculate the cross product of two vectors?

To calculate the cross product, you first determine the determinant of the 3x3 matrix formed by the two vectors and the unit vectors. Then, simplify the result to get the final vector value with x, y, and z components.

Similar threads

  • Introductory Physics Homework Help
Replies
14
Views
230
  • Introductory Physics Homework Help
Replies
8
Views
976
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
9
Views
2K
  • Introductory Physics Homework Help
Replies
12
Views
1K
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
917
  • Precalculus Mathematics Homework Help
Replies
5
Views
456
  • Introductory Physics Homework Help
Replies
2
Views
691
  • Introductory Physics Homework Help
Replies
3
Views
1K
Back
Top