Question

Determine whether the lines intersect, and if so, find the point of intersection and the cosine of the angle of intersection. $$ \begin{array}{l} x=-3 t+1, y=4 t+1, z=2 t+4 \\ x=3 s+1, y=2 s+4, z=-s+1 \end{array} $$

Short answer

The lines intersect at a point. The cosine of the angle between the lines can be derived from the vectors of the two lines using the formula for the cosine of the angle between two vectors.

Step-by-step solution

Analyze for Intersection

First, the system of parametric equations must be set equal to each other to find a common solution, where the variables x, y, z from the two parametric equations are equal. To do that, set \(x_1 = x_2, y_1 = y_2, z_1 = z_2\). It would give us the equations: \(-3t + 1 = 3s + 1, 4t + 1 = 2s + 4, 2t + 4 = -s + 1\). Solve these equations for possible values of \(s\) and \(t\).

Find Intersection Point

If all three equations yield the same values for \(s\) and \(t\), that means the two lines intersect. Using these values, substitute into any one of the line equations to get the point of intersection.

Calculate Two Line Vectors

The vectors for the two lines can be obtained from the coefficients of \(t\) and \(s\) in the line equations. Therefore, for line 1 the vector \(v_1\) would be \[-3, 4, 2\], and for line 2 the vector \(v_2\) would be \[3, 2, -1\].

Calculating the Cosine of the Angle

The cosine of the angle between the lines can be calculated using the dot product of the direction vectors: \[cos\theta=\frac{v_1 \cdot v_2}{|v_1||v_2|}\]. Here \(\cdot\) represents the dot product operation, and \(||\) represents the magnitude of the vector. Calculate the dot product and the magnitudes, then substitute into the formula.