Question

Consider three vectors: A~, B~, and C~. These vectors are related by:

A~ + B~ = 5C~
A~ - B~ = 3C~
Vector C~ has components Cx = 2 and Cy = 4. Find the x- and y-components of A~ and B~.
a) Ax = 2, Ay = 4, Bx = -1, By = 1
b) Ax = 5, Ay = 3, Bx = 2, By = 4
c) Ax = 1, Ay = 1, Bx = -2, By = -4
d) Ax = -1, Ay = -1, Bx = 2, By = -4

Answer 1

To find the x- and y-components of vectors A~ and B~, use the given equations A~ + B~ = 5C~ and A~ - B~ = 3C~. Substitute the components of vector C~ and solve the system of equations to find the x- and y-components of A~ and B~.

Step-by-step explanation:

To find the x- and y-components of vectors A~ and B~, we can use the given equations:

• A~ + B~ = 5C~
• A~ - B~ = 3C~

We know that vector C~ has components Cx = 2 and Cy = 4. We can substitute these values into the equations and solve for the components of A~ and B~.

From the first equation, we have:

• Ax + Bx = 5Cx
• Ay + By = 5Cy

Substituting Cx = 2 and Cy = 4:

• Ax + Bx = 10
• Ay + By = 20

From the second equation, we have:

• Ax - Bx = 3Cx
• Ay - By = 3Cy

Substituting Cx = 2 and Cy = 4:

• Ax - Bx = 6
• Ay - By = 12

Now, we can solve the system of equations to find the values of Ax, Ay, Bx, and By:

• Ax + Bx = 10
• Ay + By = 20
• Ax - Bx = 6
• Ay - By = 12

By adding the first and third equations, we get:

• 2Ax = 16

So, Ax = 8.

By adding the second and fourth equations, we get:

• 2Ay = 32

So, Ay = 16.

Substituting these values back into the first equation, we can find Bx:

• 8 + Bx = 10

So, Bx = 2.

Substituting these values back into the second equation, we can find By:

• 16 + By = 20

So, By = 4.

Therefore, the x- and y-components of A~ and B~ are:

• Ax = 8, Ay = 16
• Bx = 2, By = 4