Adding vectors using components – problems and solutions

1. Three vectors as shown in the figure below.

V_{1 } = 30

V_{2} = 30

V_{3 } = 40

What is the resultant vectors.

Known :

V_{1} = 30, angle between V_{1} and x axis = 30^{o}

V_{2} = 30, angle between V_{2} and x axis = 30^{o}

V_{3} = 40, angle between V_{3 }and x axis = 0^{o}

__Wanted :__ The resultant vectors

__Solution :__

V_{1x} = (V_{1})(cos 30^{o}) = (30)(0.5√3) = 15√3. Positive because this vector component points along the positive x axis (rightward).

V_{1y} = (V_{1})(sin 30^{o}) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V_{2x} = (V_{2})(cos 30^{o}) = (30)(0.5√3) = -15√3. Negative because this vector component points along the negative x axis (leftward).

V_{2y} = (V_{2})(sin 30^{o}) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V_{3x} = (V_{3})(cos 0^{o}) = (40)(1) = 40. Positive because this vector component points along the positive x axis (rightward).

V_{3y} = (V_{3})(sin 0^{o}) = (40)(0) = 0

__The components of the resultant vectors :__

V_{x} = V_{1x }– V_{2x} + V_{3x }= 15√3 – 15√3 + 40 = 40

V_{y} = V_{1y }+ V_{2y }+ V_{3y }= 15 + 15 = 30

2. Two forces perpendicular each other, F_{1 }= 12 N and F_{2} = 5 N. What is the resultant of both forces.

__Known :__

Force 1 (F_{1}) = 12 Newton

Force 2 (F_{2}) = 5 Newton

__Wanted : __The resultant vectors (ΣF)

__Solution :__

ΣF^{2} = F_{1}^{2} + F_{2}^{2 }= 12^{2 }+ 5^{2 }= 144 + 25 = 169

ΣF = √169 = 13 Newton

3. Three vectors,

V_{1 }= 30

V_{2 }= 30

V_{3} = 40

Determine the resultant vectors.

__Known :__

v_{1} = 30, makes 30^{o} about the negative x axis

v_{2} = 30, makes 30^{o} about the positive x axis

v_{3} = 40, makes 0^{o} about the positive x axis

__Wanted :__ The resultant vectors

__Solution :__

__The components of vectors :__

v_{1x} = v_{1 }cos 30^{o} = (30)(0.5√3) = -15√3 (Negative because this vector component points along the negative x axis (leftward))

v_{1y} = v_{1} sin 30^{o} = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward))

v_{2x} = v_{2 }cos 30^{o} = (30)(0.5√3) = 15√3 (Positive because this vector component points along the positive x axis (rightward))

v_{2y} = v_{2 }sin 30^{o} = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward).)

v_{3x }= v_{3 }cos 0^{o} = (40)(1) = 40 (Positive because this vector component points along the positive x axis (rightward).)

v_{3y }= v_{3} sin 0^{o} = (40)(0) = 0

__The components of the resultant vectors :__

v_{x} = – v_{1x} + v_{2x} + v_{3x} = -15√3 + 15√3 + 40 = 40

v_{y} = v_{1y} + v_{2y} + v_{3y} = 15 + 15 = 30

__The resultant vector :__

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

__Known :__

F_{1} = 3 Newton, makes 60^{o} about the positive x axis

F_{2} = 3 Newton, makes 0^{o} about the negative x axis

F_{3} = 6 Newton, makes 60^{o} about the negative y axis

__Wanted :__ The resultant vector

__Solution :__

__The components of vectors :__

F_{1x} = F_{1} cos 60^{o} = (3)(0.5) = 1.5 N (Positive because this vector component points along the positive x axis (rightward))

F_{1y} = F_{1} sin 60^{o} = (3)(0.5√3) = 1.5√3 N (Positive because this vector component points along the positive y axis (upward))

F_{2x} = F_{2} cos 0^{o} = (3)(1) = -3 N (Negative because this vector component points along the negative x axis (leftward))

F_{2y} = F_{2} sin 0^{o} = (3)(0) = 0

F_{3x} = F_{3} cos 60^{o} = (6)(0.5) = 3 N (Positive because this vector component points along the positive x axis (rightward))

F_{3y} = F_{3} sin 60^{o} = (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 :__

ΣF_{x} = F_{1x} – F_{2x} + F_{3x }= 1.5 N – 3 N + 3 N = 1.5 N

ΣF_{y} = F_{1y} + F_{2y} – F_{3y }= 1.5√3 N + 0 N – 3√3 N = -1.5√3 N

The resultant vector :

5. Two forces, F_{1 }= 15 N and F_{2} = 9 N. The angle between both vectors is 60°. What is the resultant of vectors.

__Wanted :__

Force 1 (F_{1}) = 15 Newton

Force 2 (F_{2}) = 9 Newton

Angle (θ) = 60^{o}

__Wanted:__ The resultant vector

__Solution :__

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

__Known :__

F_{1} = 20 Newton, angle between F_{1 }and x axis = 0

F_{2} = 20 Newton, angle between F_{2 }and x axis = 60

F_{3} = 24 Newton, angle between F_{3 }and x axis = 60

__Wanted :__ The resultant vector

__Solution :__

__The components of vectors :__

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

F_{1y} = (F_{1})(sin 0) = (20)(0) = 0

F_{2x} = (F_{2})(cos 60) = (20)(0.5) = -10. Negative because this vector component points along the negative x axis (leftward)

F_{2y} = (F_{2})(sin 60) = (20)(0.5√3) = 10√3. Positive because this vector component points along the positive y axis (upward)

F_{3x} = (F_{3})(cos 60) = (24)(0.5) = -12. Negative because this vector component points along the negative x axis (leftward)

F_{3y} = (F_{3})(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 :__

F_{x} = F_{1x }– F_{2x }– F_{3x }= 20 – 10 – 12 = -2

F_{y} = F_{1y }+ F_{2y }– F_{3y }= 0 + 10√3 – 12√3 = -2√3

__The resultant vector :__