Calculus 3: 3D Space

Consider the planes x + y + z = 1 and x - 2y + 3z = 1

a. Find the angle between the two planes

b. Find symmetric equations for the line of intersection of the two planes

1. Find a vector and parametric equations for the line that passes through (4,2) and is parallel to v = <-1,5>. Then find 2 other points on that line.

2. Find parametric equations for the line segment joining points P(2, -4, -1) and Q(5, 0, 7). Where does this line intersect the xy-plane?

Plot the point $B(-2,-4,-3)$ in the 3D coordinate system.

Find the line through the points (5, -2, 3) and (7, 4, 1)

Plot the point $P(2, 4, 3)$ in the 3D coordinate system.

Using a topographical map, analyze the contours to plan a route through the mountains with minimal elevation changes. Discuss the importance of this analysis in winter sports like skiing or snowshoeing.

Using the equation of a sphere, $x^2 + y^2 + z^2 = 1$, graph the sphere with the given axes.

Solve problems involving different variations of axes given a three-dimensional graph and assess the suitability of the axis variation for the right-hand rule.