Final answer:
To find the direction cosines of the line to the point (3, −5, 2), calculate the vector's magnitude, then divide the point's coordinates by this magnitude to get the cosines of the directional angles with the axes.
Step-by-step explanation:
To determine the direction cosines of the line joining the origin to the point (3,−5,2), we need to understand the concept of direction angles and their corresponding cosines. The direction cosines are the cosines of the angles a line makes with the positive directions of the coordinate axes. Given a point (x, y, z), the direction cosines l, m, and n of the line joining the origin to that point are given by:
- l = cos(α) = x / √(x² + y² + z²)
- m = cos(β) = y / √(x² + y² + z²)
- n = cos(γ) = z / √(x² + y² + z²)
In this case, for point (3, −5, 2), calculate the magnitude of the vector first:
A = √(3² + (−5)² + 2²) = √(9 + 25 + 4) = √38
Then, use the above formulas to find the direction cosines:
- l = 3 / √38
- m = −5 / √38
- n = 2 / √38
These fractions represent the cosines of the angles the line forms with the x, y, and z axes, respectively.