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 = 30^o

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

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

Wanted : The resultant vectors

Solution :

The components of vectors :

V_1x = (V₁)(cos 30^o) = (30)(0.5√3) = 15√3. Positive because this vector component points along the positive x axis (rightward).

V_1y = (V₁)(sin 30^o) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V_2x = (V₂)(cos 30^o) = (30)(0.5√3) = -15√3. Negative because this vector component points along the negative x axis (leftward).

V_2y = (V₂)(sin 30^o) = (30)(0.5) = 15. Positive because this vector component points along the positive y axis (upward).

V_3x = (V₃)(cos 0^o) = (40)(1) = 40. Positive because this vector component points along the positive x axis (rightward).

V_3y = (V₃)(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 :

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

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 30^o about the negative x axis

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

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

Wanted : The resultant vectors

Solution :

The components of vectors :

v_1x = v₁ cos 30^o = (30)(0.5√3) = -15√3 (Negative because this vector component points along the negative x axis (leftward))

v_1y = v₁ sin 30^o = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward))

v_2x = v₂ cos 30^o = (30)(0.5√3) = 15√3 (Positive because this vector component points along the positive x axis (rightward))

v_2y = v₂ sin 30^o = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward).)

v_3x = v₃ cos 0^o = (40)(1) = 40 (Positive because this vector component points along the positive x axis (rightward).)

v_3y = v₃ 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 :

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

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

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

Wanted : The resultant vector

Solution :

The components of vectors :

F_1x = F₁ cos 60^o = (3)(0.5) = 1.5 N (Positive because this vector component points along the positive x axis (rightward))

F_1y = F₁ 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 = F₂ cos 0^o = (3)(1) = -3 N (Negative because this vector component points along the negative x axis (leftward))

F_2y = F₂ sin 0^o = (3)(0) = 0

F_3x = F₃ cos 60^o = (6)(0.5) = 3 N (Positive because this vector component points along the positive x axis (rightward))

F_3y = F₃ 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, 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

Angle (θ) = 60^o

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 :

F_1x = (F₁)(cos 0) = (20)(1) = 20. Positive because this vector component points along the positive x axis (rightward)

F_1y = (F₁)(sin 0) = (20)(0) = 0

F_2x = (F₂)(cos 60) = (20)(0.5) = -10. Negative because this vector component points along the negative x axis (leftward)

F_2y = (F₂)(sin 60) = (20)(0.5√3) = 10√3. Positive because this vector component points along the positive y axis (upward)

F_3x = (F₃)(cos 60) = (24)(0.5) = -12. Negative because this vector component points along the negative x axis (leftward)

F_3y = (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 :

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 :