Find the angle between the vectors. (Round your answer to the nearest degree.)
a = ‹-4, 3›
b = ‹√7, 3›
use the dot product
a•b = |a| |b|cos Ø
cosØ = (-4√7 + 9)/(20)
Ø = 94.5º
To find the angle between two vectors, we can use the dot product formula:
a · b = |a| * |b| * cos(theta)
where a · b is the dot product of vectors a and b, |a| and |b| are the magnitudes of vectors a and b, and theta is the angle between the vectors.
First, let's find the magnitudes of the vectors:
|a| = √((-4)^2 + 3^2) = √(16 + 9) = √25 = 5
|b| = √(√7^2 + 3^2) = √(7 + 9) = √16 = 4
Now, let's find the dot product of the vectors:
a · b = (-4 * √7) + (3 * 3) = -4√7 + 9
Now we can plug these values into the dot product formula to find the angle theta:
-4√7 + 9 = 5 * 4 * cos(theta)
-4√7 + 9 = 20 * cos(theta)
cos(theta) = (-4√7 + 9) / 20
To find theta, we need to take the inverse cosine (cos^-1) of the above value:
theta ≈ cos^-1((-4√7 + 9) / 20)
Using a calculator, we find:
theta ≈ 62 degrees
Therefore, the angle between the vectors a and b is approximately 62 degrees.
To find the angle between two vectors a and b, you can use the dot product formula:
a · b = |a| |b| cos θ,
where a · b is the dot product of vectors a and b, |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between the two vectors.
First, let's find the magnitudes of vectors a and b:
|a| = √((-4)² + 3²) = √(16 + 9) = √25 = 5,
|b| = √(√7² + 3²) = √(7 + 9) = √16 = 4.
Next, let's find the dot product of a and b:
a · b = (-4)(√7) + (3)(3) = -4√7 + 9.
Now, we can plug the values into the dot product formula and solve for θ:
-4√7 + 9 = 5(4) cos θ,
-4√7 + 9 = 20 cos θ.
Dividing both sides by 20:
(-4√7 + 9) / 20 = cos θ.
Now, we can calculate the angle θ:
θ = cos⁻¹((-4√7 + 9) / 20).
Using a calculator to evaluate the expression, we find:
θ ≈ 55.7 degrees.
Therefore, the angle between vectors a and b, rounded to the nearest degree, is approximately 56 degrees.