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(1 point) Find the angle between the vectors ủ = 8ỉ – 7j and v = 5ỉ + 9j. Round to two decimal places. 0=|| radians.

User Nolandda
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1 Answer

6 votes
The answer is 2.82 radians.

To find the angle between two vectors, u and v, we can use the dot product formula:

u · v = |u| |v| cos(θ),

where u · v represents the dot product of u and v, |u| and |v| represent the magnitudes of u and v, and θ represents the angle between the vectors.

Let's calculate the dot product:

u · v = (8)(5) + (-7)(9)
= 40 - 63
= -23.

Next, let's calculate the magnitudes of u and v:

|u| = sqrt((8^2) + (-7^2))
= sqrt(64 + 49)
= sqrt(113).

|v| = sqrt((5^2) + (9^2))
= sqrt(25 + 81)
= sqrt(106).

Now, let's substitute the values into the dot product formula and solve for θ:

-23 = sqrt(113) sqrt(106) cos(θ).

Dividing both sides by sqrt(113) sqrt(106), we have:

cos(θ) = -23 / (sqrt(113) sqrt(106)).

Now we can find the angle θ by taking the inverse cosine (arccos) of the right-hand side:

θ = arccos(-23 / (sqrt(113) sqrt(106))).

Using a calculator or a trigonometric table, we can find the approximate value of θ to two decimal places:

θ ≈ 2.82 radians.

Therefore, the angle between the vectors u = 8i - 7j and v = 5i + 9j is approximately 2.82 radians.
User Anupsabraham
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