Final answer:
To find the angle between two vectors using both the dot product and direction cosine, we need to calculate the dot product of the two vectors and the magnitudes of each vector. The dot product is given by the formula: A · B = |A||B|cosθ. The angle between the vectors (2i+3j-4k) and (3i-2j+4k) is approximately 120 degrees.
Step-by-step explanation:
To find the angle between two vectors using both the dot product and direction cosine, we need to calculate the dot product of the two vectors and the magnitudes of each vector. The dot product is given by the formula: A · B = |A||B|cosθ, where A and B are the vectors, |A| and |B| are their magnitudes, and θ is the angle between them.
First, let's calculate the dot product:
A · B = (2)(3) + (3)(-2) + (-4)(4) = 6 - 6 - 16 = -16.
Next, let's calculate the magnitudes:
|A| = √(2^2 + 3^2 + (-4)^2) = √(4 + 9 + 16) = √29
|B| = √(3^2 + (-2)^2 + 4^2) = √(9 + 4 + 16) = √29
Now, let's calculate the cosine of the angle using the dot product and magnitudes:
cosθ = (A · B) / (|A||B|) = (-16) / (√29 * √29) = -16/29
Finally, let's find the angle θ using the inverse cosine:
θ = cos^(-1)((-16)/29) ≈ 120 degrees
Therefore, the angle between the two vectors is 120 degrees.