Vector Addition and the Parallelogram Rule (Grades 11-12)
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Vector Addition and the Parallelogram Rule
1.
When using the parallelogram rule to add vectors, the sum of the vectors can be either diagonal of the parallelogram formed.
- True
- False
2.
Add the vectors using the parallelogram method. [math] \ \ \ vec{u} = <<4,-6>>, \ \ \ vec{v} = <<1,-2>>[/math]

3.
Add the vectors using the parallelogram method. [math] \ \ \ \ vec{u} = <<-5,3>>, \ \ \ vec{v} = <<9,2>>[/math]

4.
In the following diagram, quadrilateral [math]FBCE[/math] is a parallelogram. Let sides [math]bar{FB}[/math], [math]bar{FE}[/math], and [math]bar{EC}[/math] represent vectors [math]vec{FB}[/math], [math]vec{FE}[/math], and [math]vec{EC}[/math]. Also, let diagonal [math]bar{FC}[/math] represent vector [math]vec{FC}[/math]. Assume that points F, E, and D are collinear. Not pictured is line segment [math]bar{CD}[/math], which is perpendicular to [math]bar{FD}[/math].
Let [math]ang BFE = theta[/math].
According to the parallelogram rule, [math]vec{FB} + vec{FE} = vec{FC}[/math]. The following questions will derive formulas for the magnitude and direction of [math]vec{FC}[/math], depending on the magnitudes and relative directions of vectors [math]vec{FB}[/math] and [math]vec{FE}[/math].
Let [math]ang BFE = theta[/math].
According to the parallelogram rule, [math]vec{FB} + vec{FE} = vec{FC}[/math]. The following questions will derive formulas for the magnitude and direction of [math]vec{FC}[/math], depending on the magnitudes and relative directions of vectors [math]vec{FB}[/math] and [math]vec{FE}[/math].

A.
Considering parallelogram [math]FBCE[/math], and that point D is collinear with points F and E, which of the following is/are true? (There may be more than one correct answer).
- [math]||vec{FE}|| + ED = FD[/math]
- [math]vec{FB} = vec{EC}[/math]
- [math]ang CED = theta[/math]
- [math]vec{FB} + vec{EC} = 2 vec{FE}[/math]
B.
Considering right triangle CDE, how is the magnitude of [math]vec{EC}[/math] related to [math]theta[/math] and the length of [math]bar{ED}?[/math]
- [math] ED = ||vec{EC}|| cos theta[/math]
- [math]ED = ||vec{EC}|| sin theta[/math]
- [math]||vec{EC}|| = ED cos theta[/math]
- [math]tan theta = ||vec{EC}|| / (ED) [/math]
C.
Considering triangle CDE, how is the magnitude of [math]vec{EC}[/math] related to the length of [math]bar{CD}[/math] and [math]theta ?[/math]
- [math]CD = ||vec{EC}|| cos theta[/math]
- [math]CD = ||vec{EC}|| sin theta[/math]
- [math] || vec{EC}|| = CD cos theta [/math]
- [math] tan theta = ||vec{EC}|| / (CD)[/math]
D.
In the following work, two reasons are not given (in steps 1 and 5). What are these reasons?
Taking the square root of both sides gives the formula for the magnitude of the sum of the vectors [math]vec{FB}[/math] and [math]vec{FE}[/math]:
[math] ||vec{FC}|| = sqrt(||vec{FE}||^2 + ||vec{FB}||^2 + 2||vec{FE}|| \ ||vec{FB}||cos theta)[/math].
[math] ||vec{FC}||^2 [/math] | [math] = [/math] | [math] FD^2 + CD^2[/math] | [math]"Step 1 "[/math] |
[math] [/math] | [math] = [/math] | [math] (||vec{FE}|| + ED)^2 + CD^2[/math] | [math] "Info from previous questions"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + 2||vec{FE}|| ED + ED^2 + CD^2 [/math] | [math] "Expanding the square"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + 2||vec{FE}|| \ ||vec{EC}||cos theta + ||vec{EC}||^2 cos^2 theta + ||vec{EC}||^2sin^2theta [/math] | [math]"Info from previous questions"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + 2||vec{FE}|| \ ||vec{EC}||cos theta + ||vec{EC}||^2 [/math] | [math]"Step 5"[/math] |
[math] [/math] | [math] = [/math] | [math] ||vec{FE}||^2 + ||vec{FB}||^2 + 2||vec{FE}|| \ ||vec{FB}||cos theta [/math] | [math]"Info from previous questions"[/math] |
Taking the square root of both sides gives the formula for the magnitude of the sum of the vectors [math]vec{FB}[/math] and [math]vec{FE}[/math]:
[math] ||vec{FC}|| = sqrt(||vec{FE}||^2 + ||vec{FB}||^2 + 2||vec{FE}|| \ ||vec{FB}||cos theta)[/math].
- Pythagorean Theorem; [math]sin^2x + cos^2x = 1[/math]
- Pythagorean Theorem; Law of Sines
- Law of Vector Addition; Law of Cosines
- Parallelogram Law; [math]cos2x = cos^2x - sin^2x[/math]
E.
Using in formation from previous questions, the value of [math]ang 2[/math] can also be determined, as given below.
[math]ang 2 = tan^{-1} ((||vec{FB}|| sin theta) / (||vec{FE}|| + ||vec{FB}|| cos theta)) [/math]
Why is the direction given in terms of the angle between [math]vec{FC}[/math] and [math]vec{FE}[/math] and not between [math]vec{FC} [/math] and [math]vec{FB}? [/math]
[math]ang 2 = tan^{-1} ((||vec{FB}|| sin theta) / (||vec{FE}|| + ||vec{FB}|| cos theta)) [/math]
Why is the direction given in terms of the angle between [math]vec{FC}[/math] and [math]vec{FE}[/math] and not between [math]vec{FC} [/math] and [math]vec{FB}? [/math]
- It has to be given relative to the horizontal vector.
- It has to be given in relation to a vector that is parallel to one of the axis.
- It doesn't matter, since the diagonals of a parallelogram bisect their respective angles.
- It doesn't matter, but it is simpler to derive the equation this way given the work already done with finding the magnitude of the vector.
5.
Given two vectors, with magnitudes 8 and 4, which have a common initial point and an angle of 37° between them, what is the resulting vector if they are added together, given as a magnitude and direction? Let the calculated angle be relative to the vector of magnitude 4. Hint: use the parallelogram rule.
- Magnitude 11.5, angle 12.1°.
- Magnitude 11.5, angle 24.9°.
- Magnitude 5.4, angle 35.9°.
- Magnitude 5.4, angle 17.1°.
6.
For vectors [math]vec{a}, vec{b}[/math], if they are placed such that their initial points are coincident, they have an angle of 60° between them. Also, [math]||vec{a}|| = 5[/math]. If these vectors are added together, the resulting vector, [math]vec{S}[/math] has a magnitude of 15 and an angle of 43.4°, relative to [math]vec{a}[/math]. What is the magnitude of [math]vec{b} ?[/math] Round answer to the nearest unit.
- [math]||vec{b}|| = 12[/math]
- [math]||vec{b}|| = 17[/math]
- [math]||vec{b}|| = 4[/math]
- Cannot be determined with the information given.
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