Suppose that a, b ∈ ℝ3 are two vectors with the following properties: b = [3, 4, 0].a · b = 10. The angle between a and b is 60◦.(a) Find the length of a.(b) Suppose in addition that a = [x, y, z], with y = 1. What are all possible values of x and z?

(a) Find the length of a:

The dot product of two vectors a and b can be expressed in terms of the angle between them and their magnitudes (lengths) as follows:

a · b = ||a|| ||b|| cos(θ)

where ||a|| and ||b|| are the magnitudes of a and b, respectively, and θ is the angle between them.

Given that a · b = 10, ||b|| = sqrt(3^2 + 4^2 + 0^2) = 5, and cos(θ) = cos(60°) = 1/2, we can substitute these values into the equation to solve for ||a||:

10 = ||a|| * 5 * 1/2 10 = 2.5 * ||a|| ||a|| = 10 / 2.5 ||a|| = 4

So, the length of a is 4.

(b) Suppose in addition that a = [x, y, z], with y = 1. What are all possible values of x and z?

We know that the magnitude of a is 4, and a = [x, 1, z]. The magnitude of a vector is given by the square root of the sum of the squares of its components, so we have:

||a|| = sqrt(x^2 + 1^2 + z^2) = 4

Squaring both sides to get rid of the square root gives:

x^2 + 1 + z^2 = 16 x^2 + z^2 = 15

This is a circle equation in the xz-plane, centered at the origin with radius sqrt(15). Therefore, x and z can be any pair of real numbers that satisfy this equation.