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You are given two vectors A= -3.00i^+7.00j^ and B⃗ =7.00ι^+2.00j^. Let the counterclockwise angles be positive.
1)What angle does A⃗ make with the ?
2)What angle does B⃗ make with the +x-axis?
3)Vector C⃗ is the sum of A⃗ and B⃗ , so C⃗ =A⃗ +B⃗ . What angle does C⃗ make with the +x-axis

Respuesta :

Luan45

1)


First we go found the value of [tex]\vec{A}[/tex]:


a² = (-3)² + 7²

a² = 9 + 49

a² = 58

a = √59

a = 7,68 u


Now we can found the angle with the +x-axis


ax = a · cos θ

3 = 7,68 · cos θ

3 / 7,68 = cos θ

0,39 = cos θ

0,39 · [tex]\mathsf{cos^{-1}}[/tex] = θ

θ = 67º <----- This is the answer


2)


First we go found the value of [tex]\vec{B}[/tex]:


b² = 7² + 2²

b² = 49 + 4

b² = 53

b = √53

b = 7,28 u


Now we can found the angle with the +x-axis


bx = b · cos θ

7 = 7,28 · cos θ

7 / 7,28 = cos θ

0,96 · [tex]\mathsf{cos^{-1}}[/tex] = θ

θ = 16º


For counterclockwise:


180 - 16 = 164º <----- This is the answer


3)


[tex]\vec{C}=\vec{A}+\vec{B}[/tex]

[tex]\vec{C}=(-3\vec{i}+7\vec{j})+(7\vec{i}+2\vec{j})[/tex]

[tex]\vec{C}=(-3\vec{i}+7\vec{i})+(7\vec{j}+2\vec{j})[/tex]

[tex] \vec{C}= 4\vec{i}+9\vec{j}[/tex]


Now we going to found the module of C vector:


c² = 9² + 4²

c² = 81 + 16

c² = 97

c = √97

c = 9,84 u


The angle is:


Cx = C · cos θ

4 = 9,84 · cos θ

4 / 9,84 = cos θ

0,40 · [tex]\mathsf{cos^{-1}}[/tex] = θ

θ = 66º


For counterclockwise:


180 - 66 = 114º <----- This is the answer


I hope you enjoy!

Ver imagen Luan45

The Pythagorean theorem can help to determine the vector's magnitude. The magnitude and angle of the given vector are 7.16 m and 67º respectively.

The magnitude of the vector:

The magnitude of the vector can be calculated by the Pythagorean theorem:

[tex]a =\sqrt { (-3)^2 + 7^2}\\\\ a = \sqrt {58}\\\\ a = 7.16 [/tex]

The angle of a resulting vector can be determined by the formula:

[tex]cos \Theta = \dfrac a{a_x} [/tex]

Here,

[tex]\Theta [/tex] - the angle of resulting vector,

So,

[tex]\rm cos \Theta = \dfrac 3{7.16}\\\\ \Theta = 67^o [/tex]

Therefore, the magnitude and angle of the given vector are 7.16 m and 67º respectively.

Learn more about vector calculation:

https://brainly.com/question/10151539