Adding vectors using components – problems and solutions

Ƙara vectors ta amfani da abubuwan haɗin kai - matsaloli da mafita

1. Three vectors as shown in the figure below.

V₁ = 30 Adding vectors using components – problems and solutions 1

V₂ = 30

V₃ = 40

What is the resultant vectors.

An sani:

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

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

V₃ = 40, angle between V₃and x axis = 0^o

Ana so: The resultant vectors

Magani:

The components of vectors :

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

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

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

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

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

V_3y = (V₃) (zunubi 0^o) = (40)(0) = 0

Abubuwan da ke cikin vectors ɗin da suka haifar:

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 :

Adding vectors using components – problems and solutions 2

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

Ka kuma duba Motsin zuciya - matsaloli da mafita

An sani:

Force 1 (F₁) = 12 Newton

Force 2 (F₂) = 5 Newton

Ana so: The resultant vectors (ΣF)

Magani:

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

ΣF = √169 = 13 Newton

3. Three vectors,

V₁= 30 Adding vectors using components – problems and solutions 3

V₂= 30

V₃ = 40

Determine the resultant vectors.

An sani:

v₁ = 30, marcas 30^o about the negative x axis

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

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

Ana so: The resultant vectors

Magani:

Abubuwan da ke cikin vectors:

v_1x = v₁kowa 30^o = (30)(0.5)√3) = -15√3 (Negative because this vector component points along the negative x axis (leftward))

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

v_2x = v₂kowa 30^o = (30)(0.5)√3) = 15√3 (Positive because this vector component points along the positive x axis (rightward))

v_2y = v₂ba 30^o = (30)(0.5) = 15 (Positive because this vector component points along the positive y axis (upward).)

v_3x= v₃kowa 0^o = (40)(1) = 40 (Positive because this vector component points along the positive x axis (rightward).)

v_3y= v₃ ba 0^o = (40)(0) = 0

Abubuwan da ke cikin vectors ɗin da suka haifar:

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

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Vector ɗin da ya haifar:

Adding vectors using components – problems and solutions 4

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

An sani:

F₁ = 3 Newton, marcas 60^o about the positive x axis Adding vectors using components – problems and solutions 5

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

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

Ana so: The resultant vector

Magani:

Abubuwan da ke cikin vectors:

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

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

F_2y = F₂ ba 0^o = (3)(0) = 0

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

F_3y = F₃ ba 60^o = (6)(0.5√3) = -3√3 N (Negative because this vector component points along the negative y axis (saukarward))

Abubuwan da ke cikin vectors ɗin da suka haifar:

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

Vector ɗin da ya haifar:

Adding vectors using components – problems and solutions 6

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

Ana so:

1arfi XNUMX (F₁) = 15 Newton

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Force 2 (F₂) = 9 Newton

Gina (θ) = 60^o

SE busca: The resultant vector

Magani:

Adding vectors using components – problems and solutions 7

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

An sani:

F₁ = 20 Newton, angle between F₁and x axis = 0 Adding vectors using components – problems and solutions 8

F₂ = 20 Newton, angle between F₂and x axis = 60

F₃ = 24 Newton, angle between F₃and x axis = 60

Ana so: The resultant vector

Magani:

Abubuwan da ke cikin vectors:

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

F_1y = (F₁)(zunubi 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 (saukarward)

Abubuwan da ke cikin vectors ɗin da suka haifar:

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

Vector ɗin da ya haifar:

Adding vectors using components – problems and solutions 9