To find the angle between two vectors using the definition of the scalar product (also known as the dot product), we can use the formula:
a · b = |a| |b| cos(θ)
Where:
- a · b is the dot product of the vectors a and b.
- |a| is the magnitude of vector a.
- |b| is the magnitude of vector b.
- θ is the angle between the vectors.
Step 1: Calculate the Dot Product
For the vectors:
a = 3i + 4j + 4k
b = 4i + 5j + 5k
The dot product can be calculated as follows:
a · b = (3)(4) + (4)(5) + (4)(5)
This results in:
a · b = 12 + 20 + 20 = 52
Step 2: Calculate the Magnitude of Each Vector
The magnitude of vector a is:
|a| = √(3² + 4² + 4²) = √(9 + 16 + 16) = √41
The magnitude of vector b is:
|b| = √(4² + 5² + 5²) = √(16 + 25 + 25) = √66
Step 3: Use the Dot Product and Magnitudes to Find cos(θ)
Now we can use the values calculated:
52 = √41 · √66 · cos(θ)
To isolate cos(θ), we rearrange the equation:
cos(θ) = 52 / (√41 · √66)
Calculating this gives us:
First, calculate the denominator:
√41 ≈ 6.403 and √66 ≈ 8.124
Thus:
|a| |b| ≈ 6.403 * 8.124 ≈ 52.01
Now substituting back:
cos(θ) ≈ 52 / 52.01 ≈ 0.9998
Step 4: Find the Angle θ
Finally, we use the arccos function to calculate the angle:
θ = arccos(0.9998)
This will yield a very small angle, approximately:
θ ≈ 1.57 degrees
In conclusion, the angle between the vectors a and b is approximately 1.57 degrees.