Math Problem Statement
Do the calculations
Solution
Let's break down the problem first to ensure we handle the calculations properly.
Part (a):
We need to determine the constant so that the angle between the line and the -plane (i.e., the xy-plane) is 45°. The equation of the line is given as:
To find , we use the fact that the angle between the direction vector of the line and the -plane (or xy-plane) can be computed using the formula for the angle between two vectors. In this case, the xy-plane has a normal vector along the -axis, i.e., .
The formula for the cosine of the angle between the line and the plane is: where is the absolute value of the z-component of the direction vector , and is the magnitude of the direction vector .
For , we know:
Using the direction vector: and
Thus, we have the equation:
Now, let's solve this equation for .The error occurred because we cannot directly solve equations involving the absolute value of the variable in this form. To proceed, we can break the equation into two cases:
- , so
- , so
Let's solve these two cases separately.The two possible solutions for are:
Thus, can either be 5 or -5 for the angle between the line and the -plane to be 45°.
Part (b):
We are asked to consider all lines through the origin that form an angle of 45° with the -plane. Then, we need to describe the points of intersection of these lines with the plane .
These lines must have direction vectors that form a 45° angle with the xy-plane, meaning their direction vector has the form , where . Since all lines pass through the origin, their parametric form is:
Now, to find the intersection with the plane , set the -coordinate of the parametric equation equal to 5:
Solving for :
Substituting this into the equations for and :
Therefore, the points of intersection are given by:
Would you like further details on these calculations, or clarification on any part? Here are 5 related questions:
- How is the angle between a line and a plane generally calculated?
- What would the solution be if the angle between the line and the plane was different (e.g., 30°)?
- How do parametric equations describe lines in 3D space?
- What role does the z-component play in the orientation of the line relative to the plane?
- Can the intersection points vary if the angle between the line and the plane is different?
Tip: When dealing with angles between lines and planes, always use the direction vector and normal vector to simplify calculations.
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Math Problem Analysis
Mathematical Concepts
Geometry
Analytic Geometry
Angles between Lines and Planes
Parametric Equations
Formulas
Cosine of the angle between a line and a plane: cos(θ) = |dz| / |d|
Magnitude of the direction vector: |d| = sqrt(a² + b² + c²)
Intersection of parametric line with plane: x1 = (5a)/c, x2 = (5b)/c
Theorems
Vector Geometry
Angle between Line and Plane
Suitable Grade Level
Grades 11-12 or College Level
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