You are given vectors A = 5.0i - 6.5j and B = -2.5i + 7.0j. A third vector C lies in the xy-plane. Vector C is perpendicular to vector A and the scalar product of C with B is 15.0. Find the x and y components to vector C.
Answer key shows x component is 8.0, y component is 6.1
I don't know how to work the problem out.
Thank you for that. I don't understand the rule for "Cy/Cx = 5/6.5; if Cx = 6.5k, then Cy = 5k." I also realize that your solution doesn't match my book's solution. Here's the step by step from the book that I'm not understanding either, perhaps someone can suggest a rule I can review for this type of problem:
The target variables are the components of C. We are given A and B. We also know A⋅C and B⋅C, and this gives us two equations in the two unknowns Cx and Cy.
A and C are perpendicular, so A⋅C = 0. AxCx + AyCy = 0, which gives 5.0Cx − 6.5Cy = 0.
B⋅C = 15.0, so −3.5Cx + 7.0Cy = 15.0
We have two equations in two unknowns Cx and Cy. Solving gives Cx = 8.0 and Cy = 6.1.
We can check that our result does give us a vector C that satisfies the two equations A⋅C = 0 and B⋅C =15.0.
To find the components of vector C, we can use the fact that vector C is perpendicular to vector A. Perpendicular vectors have a dot product of zero.
Knowing this, we can set up the following equation:
C · A = 0
Expand the dot product:
(Cx * A_x) + (Cy * A_y) = 0
Substituting the given values for A_x and A_y:
(5.0 * Cx) + (-6.5 * Cy) = 0
Simplifying this equation gives:
5.0Cx - 6.5Cy = 0
Additionally, we are given that the scalar product of vector C with vector B is 15.0. The scalar product of two vectors is given by:
C · B = Cx * B_x + Cy * B_y
Substituting the given values for B_x and B_y:
(Cx * -2.5) + (Cy * 7.0) = 15.0
Simplifying this equation gives:
-2.5Cx + 7.0Cy = 15.0
So now we have a system of two equations with two unknowns:
5.0Cx - 6.5Cy = 0 ...(1)
-2.5Cx + 7.0Cy = 15.0 ...(2)
We can solve this system of equations using any method such as substitution, elimination, or matrices. Let's use the elimination method:
Multiply equation (1) by 2:
10.0Cx - 13.0Cy = 0 ...(3)
Now add equations (2) and (3) together:
-2.5Cx + 7.0Cy + 10.0Cx - 13.0Cy = 15.0 + 0
Simplifying:
7.5Cx - 6.0Cy = 15
Rearranging the equation:
7.5Cx = 6.0Cy + 15
Dividing by 7.5:
Cx = (6.0Cy + 15) / 7.5
Simplifying further:
Cx = 0.8Cy + 2
Substitute this value of Cx in equation (1):
5.0(0.8Cy + 2) - 6.5Cy = 0
Expanding:
4.0Cy + 10 - 6.5Cy = 0
Combine like terms:
-2.5Cy + 10 = 0
Isolating Cy:
-2.5Cy = -10
Divide by -2.5:
Cy = 4
Substitute this value of Cy back into the equation for Cx:
Cx = 0.8(4) + 2
Simplifying:
Cx = 3.2 + 2
Cx = 5.2
Therefore, the x component of vector C is 5.2 and the y component of vector C is 4.
The answer key shows that the x component is 8.0 and the y component is 6.1. It seems there may have been an error in either the question or the answer key.
Just saw the flaw. -2.5 in vector B should be -3.5. This has had me stuck for too long.
slope of vector A = -6.5/5
so slope of perpendicular = 5/6.5
or
C dot A = 0
5 Cx - 6.5 Cy = 0
Cy/Cx = 5/6.5
if Cx = 6.5 k
then Cy = 5 k
C dot B = 6.5 k (-2.5) + 5 k (7) = 15
so
-16.25 k + 35 k = 15
18.75 k = 15
k = .8
so
Cx = .8 * 6.5
Cy = .8 * 5