Question

Find the angle θ (in radians) between the vectors u and D below.

u = 2/√2, 0, 2/√2
v = 1/2, √2/2√3, 1/2
Give ewact expressions for al of your answers. Use the square root symbol 'V where needed to give an wact value for your arawer, Bo sure to include parertheses where necessary e a to datinguah I (2h) trom I/ 2 K. a) First fira the dot pioduct of u and v.
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Answer 1

To find the angle between vectors u and v, we need to find the dot product of u and v and the magnitudes of u and v. The angle is then given by cos(θ) = (u · v) / (|u| · |v|).

Step-by-step explanation:

To find the angle θ between vectors u and v, we first need to find the dot product of u and v. The dot product formula is given by u · v = u1v1 + u2v2 + u3v3. Substituting the given values, we have (2/√2)(1/2) + (0)(√2/2√3) + (2/√2)(1/2). This simplifies to 1/√2 + 0 + 1/√2.

Next, we need to find the magnitudes of vectors u and v. The magnitude of vector u is given by |u| = √((2/√2)2 + 02 + (2/√2)2) = √2 + 0 + 2/√2 = √2 + 1.

The magnitude of vector v is given by |v| = √((1/2)2 + (√2/2√3)2 + (1/2)2) = 1/4 + 2/12 + 1/4 = 1/2 + 1/6 + 1/2 = 4/6 = 2/3.

Finally, the angle θ between vectors u and v is given by cos(θ) = (u · v) / (|u| · |v|). Substituting the values, we have cos(θ) = (1/√2 + 0 + 1/√2) / ((√2 + 1) · (2/3)). Simplifying this expression will give us the answer in exact form.