Given:

\[ l₁: r₁(t) = i + tj \]

\[ l₂: r₂(u) = j + u(ui + j) \]

Find the point where $ l₁ $ and $ l₂ $ intersect and find the angle between $ l₁ $ and $ l₂ $.

A) Point of intersection: (0,0), Angle between $ l₁ $ and $ l₂ $: $\pi/4$ radians

B) Point of intersection: (1,1), Angle between $ l₁ $ and $ l₂ $: $\pi/2$ radians

C) Point of intersection: (0,0), Angle between $ l₁ $ and $ l₂ $: $\pi/2$ radians

D) Point of intersection: (1,1), Angle between $ l₁ $ and $ l₂ $: $\pi/4$ radians

None of the given options correctly describe the point of intersection and angle between lines l₁ and l₂. The actual point of intersection is (1, 2) and the angle can be calculated using the dot product to be arccos(1/√5), which does not match any of the provided options.

Explanation:

To find the intersection and the angle between the lines l₁ and l₂ given by r₁(t)=i+tj and r₂ (u)=j+u(ui+j), we need to set the two equations to have the same i and j components to solve for the parameters t and u at which the intersection occurs.

From the equation of l₁, we can see that at any point on l₁, the i-component is always 1. Hence to find the intersection point, we look for a value of u such that the i-component of l₂ is also 1.

Expanding l₂, we get r₂(u) = (u²)i + (1+u)j. For the i-components to be equal, we set u² = 1 which implies u = ±1. However, since we want the j-components to also match, we need to set (1+u)=t. Only when u = 1 will we get a corresponding t = 2, giving us a perfect match for both components.

Now, the point of intersection is the value of r₁(t) or r₂(u) at these values. Thus, the point of intersection is (1, 2).

To find the angle between the lines, we calculate the dot product of their direction vectors and divide by the magnitude of each, taking the inverse cosine to find the angle. The direction vector of l₁ is j, and the direction vector of l₂ is 2i+j. The dot product is j ⋅ (2i+j) = 1. The magnitudes are √0²+1² = 1 for l₁ and √2²+1² = √5 for l₂. Thus, the cosine of the angle is 1/√5. Taking the inverse cosine, we get the angle as arccos(1/√5), which is not equal to any of the angles provided in the options. Hence, none of the options provided is correct.