Find correct to the nearest degree, the angle between p=12i - 5j and q=4i + 3j? I need solvings please?
just remember your dot product
p•q = |p|*|q|*cosθ
plug in your numbers and solve for θ
To find the angle between two vectors, we can use the dot product formula:
p · q = |p| |q| cosθ
where p · q is the dot product of vectors p and q, |p| is the magnitude of vector p, |q| is the magnitude of vector q, and θ is the angle between the two vectors.
Let's calculate each component first:
p = 12i - 5j
q = 4i + 3j
Now, let's find the magnitudes of the vectors:
|p| = sqrt((12)^2 + (-5)^2) = sqrt(144 + 25) = sqrt(169) = 13
|q| = sqrt((4)^2 + (3)^2) = sqrt(16 + 9) = sqrt(25) = 5
Next, let's calculate the dot product:
p · q = (12)(4) + (-5)(3) = 48 - 15 = 33
Now, let's substitute these values into the dot product formula:
33 = (13)(5) cosθ
Divide both sides by 65:
cosθ = 33/65
Finally, let's find the angle by taking the inverse cosine:
θ ≈ cos^(-1)(33/65) ≈ 51.06°
Therefore, the angle between p = 12i - 5j and q = 4i + 3j is approximately 51.06 degrees.
To find the angle between two vectors, you can use the dot product formula:
θ = arccos((p · q) / (|p| * |q|))
Here's how to solve it step by step:
1. Calculate the dot product of vectors p and q:
p · q = 12 * 4 + (-5) * 3 = 48 - 15 = 33
2. Determine the magnitudes (or lengths) of vectors p and q:
|p| = √(12^2 + (-5)^2) = √(144 + 25) = √169 = 13
|q| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5
3. Substitute the values into the formula:
θ = arccos(33 / (13 * 5))
4. Calculate the value of arccos(33 / (13 * 5)):
θ ≈ arccos(33 / 65)
You can use a scientific calculator or an online calculator to find the value. The answer will be in radians.
5. Convert the result from radians to degrees:
θ (in degrees) ≈ θ (in radians) * (180 / π)
π (pi) is approximately 3.14159.
By following these steps, you should be able to find the angle between vectors p and q to the nearest degree.