Question

It is known that the vectors u=[1,0,2] and v=[3,1,1] in R3. a. Find a vector that is orthogonal to u. b. Determine the length of u and v. c. Determine the value of the cosine of the angle formed by u

Answer 1

A vector orthogonal to u=[1,0,2] is [0,1,0]. The lengths of u and v=[3,1,1] are √5 and √11, respectively. The cosine of the angle between u and v can be calculated using the dot product and the magnitudes of u and v.

Step-by-step explanation:

To address the student's mathematics question regarding vectors in R3:

1. A vector that is orthogonal to u=[1,0,2] can be found by ensuring the dot product with u equals zero. One example is [0,1,0] because the dot product (1⋅0 + 0⋅1 + 2⋅0) equals zero.

2. The length or magnitude of vectors u and v=[3,1,1] can be calculated using the formula √(x² + y² + z²). Therefore, the length of u is √(1²+0²+2²) = √5 and the length of v is √(3²+1²+1²) = √11.

3. The cosine of the angle between u and v is given by the dot product of u and v divided by the product of their magnitudes, cos(θ) = (u ⋅ v) / (|u||v|), which calculates to (3⋅1 + 1⋅0 + 2⋅1) / (√5√11).