Final answer:
To find the angle between vectors â and ëb, compute their scalar (dot) product and magnitudes, then use the formula cos θ = (â · ëb) / (A × B) and find the inverse cosine of the result.
Step-by-step explanation:
To calculate the angle between two vectors â and ëb using their scalar components, first compute the scalar product (dot product) of the vectors, and then find the magnitudes of each vector. The dot product of vectors â and ëb is given by â · ëb = axbx + ayby + azbz, where ax, ay, az and bx, by, bz are the respective components of vectors â and ëb.
For vectors â = 5.0î + 5.0ơ + 4.0âk and ëb = 6.0î + 3.0ơ + 5.0âk, the dot product is:
â · ëb = (5.0)(6.0) + (5.0)(3.0) + (4.0)(5.0) = 30 + 15 + 20 = 65
To find the magnitudes of vector â and ëb we use the formula A = √(ax² + ay² + az²) and similar for B. For â this is A = √(5.0² + 5.0² + 4.0²) = √(25 + 25 + 16) = √66 and for ëb it is B = √(6.0² + 3.0² + 5.0²) = √(36 + 9 + 25) = √70.
Substituting the dot product and the magnitudes into the equation for cos θ, we have cos θ = â · ëb / (A × B), so cos θ = 65 / (√66 × √70). Finally, the angle between the vectors is obtained by taking the inverse cosine of this expression, θ = cos⁻¹(65 / (√66 × √70)).