# Adding vectors using components – problems and solutions

Adding vectors using components – problems and solutions

1. Three vectors as shown in the figure below.

V1 = 30

V2 = 30

V3 = 40

What is the resultant vectors.

Known :

V1 = 30, angle between V1 and x axis = 30o

V2 = 30, angle between V2 and x axis = 30o

V3 = 40, angle between V3 and x axis = 0o

Wanted : The resultant vectors

Solution :

The components of vectors :

V1x = (V1)(cos 30o) = (30)(0.5√3) = 15√3. Positive because this vector component points along the positive x axis (rightward).

V1y = (V1)(sin 30o) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V2x = (V2)(cos 30o) = (30)(0.5√3) = -15√3. Negative because this vector component points along the negative x axis (leftward).

V2y = (V2)(sin 30o) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V3x = (V3)(cos 0o) = (40)(1) = 40. Positive because this vector component points along the positive x axis (rightward).

V3y = (V3)(sin 0o) = (40)(0) = 0

The components of the resultant vectors :

Vx = V1x – V2x + V3x = 15√3 – 15√3 + 40 = 40

Vy = V1y + V2y + V3y = 15 + 15 = 30

The resultant vector :

2. Two forces perpendicular each other, F1 = 12 N and F2 = 5 N. What is the resultant of both forces.

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Known :

Force 1 (F1) = 12 Newton

Force 2 (F2) = 5 Newton

Wanted : The resultant vectors (ΣF)

Solution :

ΣF2 = F12 + F22 = 122 + 52 = 144 + 25 = 169

ΣF = 169 = 13 Newton

3. Three vectors,

V1 = 30

V2 = 30

V3 = 40

Determine the resultant vectors.

Known :

v1 = 30, makes 30o about the negative x axis

v2 = 30, makes 30o about the positive x axis

v3 = 40, makes 0o about the positive x axis

Wanted : The resultant vectors

Solution :

The components of vectors :

v1x = v1 cos 30o = (30)(0.53) = -153 (Negative because this vector component points along the negative x axis (leftward))

v1y = v1 sin 30o = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward))

v2x = v2 cos 30o = (30)(0.53) = 153 (Positive because this vector component points along the positive x axis (rightward))

v2y = v2 sin 30o = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward).)

v3x = v3 cos 0o = (40)(1) = 40 (Positive because this vector component points along the positive x axis (rightward).)

v3y = v3 sin 0o = (40)(0) = 0

The components of the resultant vectors :

vx = – v1x + v2x + v3x = -153 + 153 + 40 = 40

vy = v1y + v2y + v3y = 15 + 15 = 30

The resultant vector :

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4. What is the resultant of three vectors as shown in figure below :

Known :

F1 = 3 Newton, makes 60o about the positive x axis

F2 = 3 Newton, makes 0o about the negative x axis

F3 = 6 Newton, makes 60o about the negative y axis

Wanted : The resultant vector

Solution :

The components of vectors :

F1x = F1 cos 60o = (3)(0.5) = 1.5 N (Positive because this vector component points along the positive x axis (rightward))

F1y = F1 sin 60o = (3)(0.5√3) = 1.5√3 N (Positive because this vector component points along the positive y axis (upward))

F2x = F2 cos 0o = (3)(1) = -3 N (Negative because this vector component points along the negative x axis (leftward))

F2y = F2 sin 0o = (3)(0) = 0

F3x = F3 cos 60o = (6)(0.5) = 3 N (Positive because this vector component points along the positive x axis (rightward))

F3y = F3 sin 60o = (6)(0.5√3) = -3√3 N (Negative because this vector component points along the negative y axis (downward))

The components of the resultant vectors :

ΣFx = F1x – F2x + F3x = 1.5 N – 3 N + 3 N = 1.5 N

ΣFy = F1y + F2y – F3y = 1.5√3 N + 0 N – 3√3 N = -1.5√3 N

The resultant vector :

5. Two forces, F1 = 15 N and F2 = 9 N. The angle between both vectors is 60°. What is the resultant of vectors.

Wanted :

Force 1 (F1) = 15 Newton

Force 2 (F2) = 9 Newton

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Angle (θ) = 60o

Wanted: The resultant vector

Solution :

6. What is the resultant of three vectors as shown in the figure below?

Known :

F1 = 20 Newton, angle between F1 and x axis = 0

F2 = 20 Newton, angle between F2 and x axis = 60

F3 = 24 Newton, angle between F3 and x axis = 60

Wanted : The resultant vector

Solution :

The components of vectors :

F1x = (F1)(cos 0) = (20)(1) = 20. Positive because this vector component points along the positive x axis (rightward)

F1y = (F1)(sin 0) = (20)(0) = 0

F2x = (F2)(cos 60) = (20)(0.5) = -10. Negative because this vector component points along the negative x axis (leftward)

F2y = (F2)(sin 60) = (20)(0.5√3) = 10√3. Positive because this vector component points along the positive y axis (upward)

F3x = (F3)(cos 60) = (24)(0.5) = -12. Negative because this vector component points along the negative x axis (leftward)

F3y = (F3)(sin 60) = (24)(0.5√3) = -12√3. Negative because this vector component points along the negative y axis (downward)

The components of the resultant vectors :

Fx = F1x – F2x – F3x = 20 – 10 – 12 = -2

Fy = F1y + F2y – F3y = 0 + 10√3 – 12√3 = -2√3

The resultant vector :

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