Final answer:
To find the angle between the velocity and acceleration vectors, we use the dot product formula and calculate the magnitudes of both vectors. We then find the cosine of the angle and use the inverse cosine function to find the angle itself.
Step-by-step explanation:
To determine the angle between the velocity vector v = −1i + (−4j) + 6k m/s and the acceleration vector a = 1i + (−2j) + 10k m/s², we can use the dot product formula:
v · a = |v||a|cosθ,
where θ is the angle between the vectors, |v| is the magnitude of v, and |a| is the magnitude of a. The dot product v · a is calculated as:
(−1)(1) + (−4)(−2) + (6)(10) = −1 + 8 + 60 = 67,
and the magnitudes are calculated using:
|v| = √((−1)² + (−4)² + 6²) = √(1 + 16 + 36) = √53,
|a| = √(1² + (−2)² + 10²) = √(1 + 4 + 100) = √105.
To find the angle θ:
cosθ = (67) / (√53 * √105),
θ = cos¹(67 / (√53 * √105)).
The final step is to calculate the angle using a calculator, ensuring the calculator is set to the correct mode (degrees or radians) as required by the context of the problem. This calculation will give us the angle θ between the velocity and acceleration.