The angle between two vectors can be calculated using the dot product formula:

a.b = |a||b|cosθ

where:
- a.b is the dot product of vectors a and b
- |a| and |b| are the magnitudes of vectors a and b
- θ is the angle between a and b

First, let's calculate the dot product of vectors a and b:

a.b = (2*-4) + (3*1) + (-1*5) = -8 + 3 - 5 = -10

Next, let's calculate the magnitudes of vectors a and b:

|a| = sqrt((2^2) + (3^2) + (-1^2)) = sqrt(4 + 9 + 1) = sqrt(14)

|b| = sqrt((-4^2) + (1^2) + (5^2)) = sqrt(16 + 1 + 25) = sqrt(42)

Now, we can substitute these values into the dot product formula and solve for θ:

-10 = sqrt(14)*sqrt(42)*cosθ

cosθ = -10 / (sqrt(14)*sqrt(42))

θ = cos^-1(-10 / (sqrt(14)*sqrt(42)))

Using a calculator, we find that θ ≈ 120.52 degrees.

Question

The angle between two vectors can be calculated using the dot product formula:

a.b = |a||b|cosθ

where:
- a.b is the dot product of vectors a and b
- |a| and |b| are the magnitudes of vectors a and b
- θ is the angle between a and b

First, let's calculate the dot product of vectors a and b:

a.b = (2*-4) + (3*1) + (-1*5) = -8 + 3 - 5 = -10

Next, let's calculate the magnitudes of vectors a and b:

|a| = sqrt((2^2) + (3^2) + (-1^2)) = sqrt(4 + 9 + 1) = sqrt(14)

|b| = sqrt((-4^2) + (1^2) + (5^2)) = sqrt(16 + 1 + 25) = sqrt(42)

Now, we can substitute these values into the dot product formula and solve for θ:

-10 = sqrt(14)*sqrt(42)*cosθ

cosθ = -10 / (sqrt(14)*sqrt(42))

θ = cos^-1(-10 / (sqrt(14)*sqrt(42)))

Using a calculator, we find that θ ≈ 120.52 degrees.

Knowee AI · Accepted Answer