Math Problem Statement

Find the distance from the point (8, 0, 0) to the line: x = 8 + t, y = 2t, z = 6√5 + 2t.

Solution

The problem asks to find the distance from the point (8,0,0)(8, 0, 0) to the line given by the parametric equations:

x=8+t,y=2t,z=65+2t.x = 8 + t, \quad y = 2t, \quad z = 6\sqrt{5} + 2t.

Step 1: Identify the direction vector of the line.

From the parametric equations, the direction vector v\mathbf{v} of the line is the coefficients of tt in each equation:

v=(1,2,2).\mathbf{v} = (1, 2, 2).

Step 2: Define the point on the line.

At t=0t = 0, the point on the line is:

P0=(8,0,65).P_0 = (8, 0, 6\sqrt{5}).

Step 3: Use the formula for the distance from a point to a line.

The formula for the distance from a point P1P_1 to a line passing through a point P0P_0 with direction vector v\mathbf{v} is:

d=AP0×vv,d = \frac{\left| \mathbf{AP_0} \times \mathbf{v} \right|}{|\mathbf{v}|},

where AP0\mathbf{AP_0} is the vector from the point P1P_1 to the point P0P_0.

  • P1=(8,0,0)P_1 = (8, 0, 0) (the point we are calculating the distance from),
  • P0=(8,0,65)P_0 = (8, 0, 6\sqrt{5}),
  • v=(1,2,2)\mathbf{v} = (1, 2, 2).

Step 4: Compute the vectors and the cross product.

Now, I'll calculate the distance using these parameters.The distance from the point (8,0,0)(8, 0, 0) to the given line is approximately 1010 units. This matches the correct answer from the choices provided.

Would you like further details on any step?

Here are 5 related questions for further practice:

  1. How do you find the shortest distance between two skew lines?
  2. What is the distance from a point to a plane?
  3. How do you derive the equation of a plane given three points?
  4. How is the dot product used in finding angles between vectors?
  5. What is the geometric interpretation of the cross product?

Tip: In 3D geometry, the direction vector of a line plays a crucial role in determining distances and angles between lines and planes.

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Math Problem Analysis

Mathematical Concepts

Distance in 3D
Vector Cross Product
Line Parametric Equations

Formulas

Distance from a point to a line: d = |AP × v| / |v|
Direction vector of the line: v = (1, 2, 2)
Cross product and magnitude

Theorems

Distance from a Point to a Line in 3D

Suitable Grade Level

University Level (Calculus and 3D Geometry)