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

1)

First we go found the value of \(\vec{A}\):

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 · \(\mathsf{cos^{-1}}\) = θ

θ = 67º <----- This is the answer

2)

First we go found the value of \(\vec{B}\):

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 · \(\mathsf{cos^{-1}}\) = θ

θ = 16º

For counterclockwise:

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

3)

\(\vec{C}=\vec{A}+\vec{B}\)

\(\vec{C}=(-3\vec{i}+7\vec{j})+(7\vec{i}+2\vec{j})\)

\(\vec{C}=(-3\vec{i}+7\vec{i})+(7\vec{j}+2\vec{j})\)

\(\vec{C}= 4\vec{i}+9\vec{j}\)

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 · \(\mathsf{cos^{-1}}\) = θ

θ = 66º

For counterclockwise:

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

I hope you enjoy!

priyankatutor priyankatutor · Answer 2 · 2022-01-22T06:12:21-05:00

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:

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

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

\(cos \Theta = \dfrac a{a_x}\)

Here,

\(\Theta\) - the angle of resulting vector,

So,

\(\rm cos \Theta = \dfrac 3{7.16}\\\\ \Theta = 67^o\)

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