Question

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Given the three points A(5,2,-3), B(2,-2,-1), C(2,-4,3), let:
- L1 be the line through A parallel to ([ -1; -5; 5 ]),
- P be the plane through B(-1 3 -4)
- E be the point of intersection of L1 and P,
- S be the sphere through A, B, C, E, - F be the centre of S,
- L2 be the line through C and F. Using the geom 3 d package, or otherwise:
(i) Find a decimal approximation to the angle between L1 and

Answer 1

The angle between a line and a plane is the complementary angle to that formed by the line and the plane's normal vector. To find it, compute the angle between the line and a normal to the plane using the dot product, and subtract that angle from 90 degrees.

Step-by-step explanation:

The angle between a line and a plane can be defined using the complementary angle to the one formed by the line and the normal (perpendicular) to the plane. If we consider the line L1 and a line perpendicular to the plane P, which we can call OP, the angle of interest would be (90° - X), where X is the angle between L1 and OP.

To find the angle between L1 and the plane, we would need to compute the angle between L1 and a normal vector to P. The normal vector can be found using the cross product of two non-parallel vectors lying on the plane. After finding the normal vector, one can use the dot product to find the angle between L1 and this normal, and then take the complementary angle to get the angle between L1 and the plane.